


















































































r 








V 

























The Coal Miner’s 
Handbook 


A HANDY REFERENCE BOOK 

FOR 

Coal Miners, Pit Bosses, Fire Bosses, Foremen, 
Superintendents, Managers, Engineers, 
and All Persons Interested in the 
' Subject of Coal Mining 


BY 

International Correspondence Schools 

H 

SCRANTON, PA. 


1st Edition, 6th Thousand, 1st Impression 


© > 

® ) ) 

' 

SCRANTON, PA. 

International Textbook Company 





r N ? o i> 

. 1(0 


f 


Copyright, 1913, by 
International Textbook Company 
Copyright in Great Britain 
All Rights Reserved 


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4 ^ /#) 5 ^ 



24566 


©CI.A347221 






PREFACE 

This Handbook is intended for all who are 
interested in coal mining and for all who are 
employed in and about the coal mines. While 
the treatment of some of the subjects included 
is necessarily brief, we have striven to anticipate 
the daily wants of the user and to give him, in 
the manner best suited to his needs, the informa¬ 
tion he desires. The breaker boy, the driver, the 
helper will find many useful hints to help him 
in his work and to assist him in securing advance¬ 
ment. 

From a vast number of reference tables and 
formulas only the best have been selected and 
incorporated, and these have been thoroughly 
explained. This feature in itself should result 
in a great saving of time and in preventing the 
selection of the wrong table or formula. The 
subjects of surveying, use and care of wire ropes 
in connection with hoisting and haulage, electric¬ 
ity, opening of mines, timbering, methods of 
working, and of ventilation have received special 
attention. Safety appliances, which include elec¬ 
tric signaling devices and safety lamps, are 
treated in detail, as is also the care and use of 


m 


IV 


PREFACE 


explosives. Considerable space is also devoted to 
the treatment of persons injured in and about the 
mines and those overcome with mine gases. 

The man employed on the surface will find 
recorded many useful facts dealing with surface 
plants, considerable space being devoted to dams, 
pumping, steam, preparation of coal, etc. 

This little book should satisfy a want long 
existing in the coal-mining industry for a ready 
pocket reference containing information that any- 
once can use in making calculations and settling 
questions. 

International Correspondence Schools 
March 1, 1913 


INDEX 


A 

Acetylene mine lamps, 256. 

Acid waters, Pumps for, 129. 

Afterdamp, 253. 

Air brattice, 288. 
bridge, 288. 

column, Calculation of mo¬ 
tive column or, 281. 
Composition and measure¬ 
ment of, 247. 
Compressed, 144. 

-current. Natural division 
of, 271. 

-current. Proportional divi¬ 
sion of, 272. 

-current, Splitting of, 270. 
-current, Velocity of, 260. 
currents. Conducting, 287. 
Equal splits of, 271. 
in mine ventilation. Dis¬ 
tribution of, 270. 
in pipes, Transmission of, 
144. 

Proportionate division of, 
277. 

required for ventilation. 
Quantity of, 259. 

Alabama methods, 222. 

Alternating-current circuits, 
Power in, 158. 

-current generators, 149. 
-current motors, 149. 

Aluminum wire, 162. 

American coals, Analyses and 
heating values of, 134. 

Analyses and heating values 
of American coals, 134. 

Aneroid barometer, 248. 

Angles or arcs, Measure of, 2. 

Anthracite coal, 130. 

Handling of, 313. 

Methods of mining, 229. 


Anthracite or hard coal, 130. 

Revolving screen mesh for, 
311. 

screens, Duty of, 311. 

will run, Table of pitch at 
which, 314. 

Arc for any radius, To bend 
rails to proper, 300. 

lamps, 165. 

Arcs, Measure of angles or, 2. 

Area of cut timber. Table for, 
9. 

of round timber. Table for, 

8 . 

of tract of land. Determin¬ 
ing, 84. 

Atmospheric pressure, Calcu¬ 
lation of, 249. 

Avoirdupois weight, 2. 

Axle, Wheel and, 93. 

Azimuth course, 83. 

B 

Barometer, Aneroid, 248. 

Mercurial, 248. 

Bars, Platform, 309. 

Batteries, 165. 

Battery, Double-chute, 235. 

Single-chute, 234. 

working, 232. 

Beam, Breaking strength of a, 

102 . 

Beams, Tables for bending 
moment and deflection 
of, 100. 

Bell wiring, 166. 

Bending moment and deflec¬ 
tion of beams, Table for, 
100 . 

Biram ventilator, 284. 

Bituminous, or soft coal, 130. 

Blackdamp, 251. 


v 






VI 


INDEX 


Blasting by electricity, 244. 

Blowers, Force fans and, 283. 

Board measure,Timber and, 8. 

Boiler incrustation, 139. 

Boilers, Horsepower of, 137. 

Steam, 137. 

Bord-and-pillar method of 
mining, 202. 

Box regulator, 272. 

regulator, Calculation of 
pressure for, 273. 

Brattice, Air, 288. 

Breaking strength of a beam, 

102 . 

strength of columns, 102. 

Breasts, Buggy, 215. 

Chute, 216. 

Bridge, Air, 288. 

Briquetting, 315. 

British thermal unit, 133. 

Brown coal, or lignite, 131. 

Bucket, Sinking, 177. 

Buggy breasts, 215. 

C 

Calculation of atmospheric 
pressure, 249. 

of mine resistance, 263. 

of motive column or air 
column, 281. 

of power, or units of work 
per minute, 264. 

of pressure for box reg¬ 
ulator, 273. 

of tension of haulage rope, 
295. 

of ventilating pressure in 
furnace ventilation, 280. 

Cannel coal, 131. 

Canvas doors, 288. 

Capacity, Metric measures 
of, 4. 

of pumps and horsepower 
required to raise water, 
127. 

Capell fan, 287. 

Centrifugal fans, 282. 

fans, Types of, 284. 

Chain, 78. 

cables, Resistance and 
proof tests of wrought- 
iron, 117. 


j Chains, 115. 

Charge, Firing a, 242. 

Tamping a, 242. 

Charging explosives, 241. 
Chokedamp, 251. 

Chute breasts, 216. 

Circuit, Series, 150. 

Circuits, Electric, 150. 
Electric-haulage, 153. 
Parallel, 151. 

Power in alternating-cur¬ 
rent, 158. 

Power in direct-current, 
158. 

Protection of, 153. 
Classifying apparatus, Sizing 
and,309. 

Cleaning safety lamps, 256. 
Clearfield region method of 
mining, 220. 

Clinometer, 79. 

Closed work method of mi¬ 
ning, 201. 

Coal, Anthracite, 130. 
Bituminous, 130. 

Cannel, 131. 

-crushing machinery, 307. 
Formations likely to con¬ 
tain, 168. 

Methods of working, 200. 
or bedded materials, 170. 
Removal of sulphur from, 
313. 

seam. Determining con¬ 
tents of, 84. 
Semibituminous, 130. 
Splint, 131. 
storage, 214. 

Systems of working, 202. 
The preparation of, 307. 
Coals, Classification of, 130. 
Coking. 131. 

Composition of, 131. 

Table of space occupied by 
* 2,000 pounds of various, 

315. 

Coefficient of elasticity. Table 
for, 100. 
of friction, 97. 

Coking coals, 131. 

Columns, Breaking strength 
of, 102. 



INDEX 


Vll 


Columns, Constant used in 
formula for, 103. 

Combustion, Spontaneous, 
214. 

Common logarithms of num¬ 
bers, Table for, 33. 

Compass, The, 76. 

Composition and measure¬ 
ment of air, 247. 
of coals, 131. 
of fuels, Table of, 136. 

Compressed air, 144. 

-air locomotive, 298. 

Conducting air currents, 287. 

Connellsville region method 
of mining, 218. 

Construction of dams in 
mines, 120. 

Contents of coal seam, De¬ 
termining, 84. 

Control of roof pressure, 
207. 

Conversion table for English 
measures into metric, 6. 
table for metric measures 
into English, 7. 
tables, Metric, 5. 

Copper wire, Properties of, 
161. 

Cornish pumps, 126. 

Cribs, 207. 

Crushers, Selection of, 307. 

Crushing machinery, Coal, 
307. 

Cubic measure, 2. 

Culm, Flushing of, 239. 

Current, Definition of, 155. 
Horsepower or power of, 
260. 

Velocity of air, 260. 

Currents, Conducting air, 
287. 

Measurement of ventila¬ 
ting, 261. 

Curves, 300. 

in a mine, Laying out, 90. 

Cuttimber.Tablefor area of, 9. 

D 

Dams, 120. 

Debris, 125. 

Earth, 124. 


Dams in mines. Construction 
of, 120. 

Masonry, 125. 

Stone, 124. 

Wooden, 124. 

Data, Wire, 159. 

Debris dams, 125. 

Decimal equivalents, 12. 
equivalents of parts of 1 
inch, Table for, 9. 

Decimals of a foot for each 
arj-inch, Table for, 10. 

Depth of shafts, Finding, 249. 

Detaching hooks, 294. 

Determining area of tract of 
land, 84. 

contents of coal seam, 84. 

Direct-current circuits, Power 
in, 158. 

-current generators, 147. 
-current motors, 147. 
-current motors. Efficien¬ 
cies of, 162. 

Disk fan, 282. 

Distance between centers of 
breasts or chambers. 
Table of, 212. 

Distribution of air in mine 
ventilation, 270. 

Door regulator, 272. 

regulator, Side opening for, 
274. 

Doors, Canvas, 288. 

Double-chute battery, 235. 

Drainage and ventilation 
during sinking, 178. 

Drawing pillars, 211. 

Drill holes. Arrangement of, 
245. 

Driving the gangway, 179. 

Drums on life of wire ropes, 
Effects of sheaves or, 
110 . 

Dry measure, 2. 

Dunn’s tables of size of room 
pillars for various depths, 
211 . 

Duplex pumps, Simple and, 
126. 

Duty of anthracite screens, 
311. 

Dynamite, Thawing, 245. 




Vlll 


INDEX 


E 

Earth dams, 124. 

Effects of sheaves or drums on 
life of wire ropes, 110. 

Efficiency of steam at various 
pressures, 138. 

Elasticity, Table for coeffi¬ 
cient of, 100. 

Electric apparatus in fire¬ 
damp, 154. 
circuits, 150. 
exploder, 243. 
generators and motors, 147. 
-haulage circuits, 153. 
mine lamps, 257. 

-mining locomotive, 298. 
signaling, 165. 

Electrical calculations, 155. 
units, 155. 

units, Mechanical equiv¬ 
alents of, 156. 

Electricity, 147. 

Blasting by, 244. 

Electromotive force, Defini¬ 
tion of, 155. 

Elevators, Miscellaneous 
forms of water, 128. 

Elongation and shortening 
under stress, 102. 

Endless-rope haulage, Deter¬ 
mination of friction pull 
on, 296. 

-rope system, 296. 

Engine, Sinking, 178. 

Engines, Steam, 142. 

English, Conversion table for 
metric measure into, 7. 
measures into metric, Con¬ 
version table for, 6. 

Entries, Number of, 208. 

Equal splits of air, 271. 

Equivalent orifice, The, 264. 

Examples in solutions of tri¬ 
angles, 60. 

Exhaust fans, 283. 

Exploder, Electric, 243. 

Exploration by drilling or 
bore holes, 171. 

Explosives, 240. 

Charging, 241. 

Permissible, 240. 

Explosions, Mine, 258. 


F 

Factors of safety, Table for, 
99. 

Fan, Capell, 287. 

Disk, 282. 

Nasmyth, 284. 

Position of, 283. 

Fans and blowers, Force, 
283 

Centrifugal, 282. 

Exhaust, 283. 

Types of centrifugal, 284. 

Fastenings, Wire-rope, 113. 

Faults, Schmidt’s law of, 
171. 

Firedamp, 252. 

Electric apparatus in, 154. 

Firing a charge. 242. 

Flattened-strand ropes, 104. 

Flow of air in pipes, Table of 
loss of pressure by, 146. 
of water through pipes, 
119. 

Flushing of culm, 239. 

Force fans and blowers, 
283. 

Formations likely to contain 
coal, 168. 

Formula, Rankine-Gordon, 
27. 

Formulas, 25. 

for lamp wiring, 163. 
for primary splits, Table of 
ventilating, 275. 
for secondary splits, Table 
of ventilating, 276. 
Heating, 133. 

Splitting, 275. 

Table of ventilation, 267. 
Wire, 162. 

Friction and lubrication, 97. 
Coefficient of, 97. 
pull on endless-rope haul¬ 
age, Determination of, 
296. 

Fuels, 130. 

Table of composition of, 
136. 

Furnace ventilation, 280. 
ventilation, Calculation of 
ventilating pressure in, 
280. 



INDEX 


IX 


G 

Gangway, Driving the, 179. 
Gas by lamp flame, Testing 
for, 253. 

Treatment of persons over¬ 
come by, 318. 

Gases found in mines, 250. 

Table of mine, 254. 
Gathering locomotives, 297. 
Gauge of track, 300. 

Wire, 160. 

Generators, Alternating-cur¬ 
rent, 149. 

and motors. Electric, 147. 
Direct-current, 147. 
Geometry, 52. 

George Creek District, Md., 
method of mining, 223. 
Grades. 299. 

Gravity planes, 294. 

Guibal ventilator, 286. 

H 

Hard, or anthracite coal, 130. 
Haulage, 294. 

and transmission ropes, 
Table of hoisting, 106. 
Motor, 297. 

Rope, 295. 

rope, Calculation of ten¬ 
sion of, 295. 

Speed of, 299. 

Head-frames, 293. 

-frames or head-gears, Tim¬ 
ber, 200. 

-frames, Sinking, 177. 
Head-gears, Timber head- 
frames or, 200. 
Head-sheaves, 293. 

Heating formulas, 133. 

values of American coals, 
Analyses and, 134. 
High-pressure steam, 138. 
Hoisting, 288. 

haulage and transmission 
ropes, Table of, 106. 
Koepe system of, 289. 
Power used for, 291. 
Problems in, 291. 
ropes, Starting strain on, 
108. 

Whiting system of, 290. 


Horsepower of boilers; 137. 
of manila rope, 117. 
or power of current, 260. 
Hydrogen sulphide, 252. 
Hydromechanics, 118. 
Hydrostatics, 118. 

I 

Illuminating power of safety 
lamps, 256. 

Incandescent-lamp ratings, 
163. 

Inclined planes, 95. 
roads, 297. 

Incrustation, Boiler, 139. 
Prevention of, 140. 
remedies, 140. 

Indiana coal-mining method, 
223. 

Induction motors. Efficien¬ 
cies of, 163. 

Injured persons. Treatment 
of, 316. 

Instruments, Surveying, 76. 
Iowa coal-mining method, 
223. 

Iron and steel supports, 195. 

J 

Jigs, 312. 

K 

Keeping surveying notes, 87. 
Koepe system of hoisting, 
289. 

L 

Landings, 193. 

Lamp wiring. Formulas for, 
163. 

Lamps, Acetylene mine, 256. 
Arc, 165. 

Cleaning safety, 256. 
Electric mine, 257. 
Illuminating power of 
safety, 256. 

Locking safety, 256. 

Oil for safety, 255. 

Safety, 254. 

Table of light given by 
satety, 257. 

Types of safety, 255. 
Wiring calculations for,. 
163. 





X 


INDEX 


Laws, Mine ventilation, 270. 

Lay of rope, 104. 

Laying out curves in a mine, 
90. 

Length, Metric measure of, 4. 

Levers, 92. 

Light given by safety lamps, 
Table of, 257. 

Lignite, or brown coal, 131. 

Linear measure, 1. 

Liquid measure, 3. 

Locking safety lamps, 256. 

Locomotive, Compressed-air, 
298. 

Electric-mining, 298. 

Locomotives, Gathering, 297. 

Logarithms, 28. 

of numbers, Table of com¬ 
mon, 33. 

Longwall face, Timbering, 
207. 

method, Modifications of, 
229. 

method of mining, 204. 

Long-ton table, 2. 

Lubricants for different pur¬ 
poses, Best, 97. 

Lubrication, Friction and, 97. 

of ropes, 112. 

M 

Machine mining, 246. 

Machinery, Coal crushing, 
307. 

Manila rope, Horsepower of, 
117. 

Masonry dams, 125. 

Material required for 1,000 ft. 
and for 1 M. of single 
track, 302. 

required for single-track 
road, Table of, 305. 

Materials, Strength of, 98. 

Meandering, Definition of, 
81. 

Measure, Cubic, 2. 

Dry, 2. 

Linear, 1. 

Liquid, 3. 

of angles or arcs, 2. 

of length, Metric, 4. 

Square, 1. 


Measurement of air, Com¬ 
position and, 247. 
of ventilating currents, 261. 
of pressure, 261. 
of temperature, 263. 
of velocity, 261. 

Measures and weights, 1. 
of capacity, Metric, 4. 
of surface, Metric, 4. 
of volume, Metric, 4. 
of weight, Metric, 5. 

Mechanical equivalents of 
electrical units, 156. 
ventilators, 281. 

Mechanics, Elements of, 92. 

Mensuration, 54. 

M. E. P., Determination of, 
143. 

Mercurial barometer, 248. 

Mercury column, Water col¬ 
umn corresponding to 
any,249. 

Mesh for anthracite, Revolv¬ 
ing screen, 311. 
for shaking screens. Table 
of, 310. 

Methane, 250. 

Methods of mining anthra¬ 
cite, 229. 

of working coal, 200. 

Metric, Conversion table for 
English measures into, 6. 
measures into English, Con¬ 
version table for, 7. 
system, 3. 

Mine corps, 90. 
explosions, 258. 
gases, Table of, 254. 
lamps, Acetylene, 256. 
lamps, Electric, 257. 
Opening a, 172. 

Potential factor of a, 264. 
resistance, 260. 
resistance, Calculation of, 
263. 

roads and tracks, 299. 
timber and timbering, 180. 
timbering and underground 
supports. Forms of, 183. 
timbering, Joints in, 182. 
ventilation. Distribution of 
air in, 270. 



INDEX 


xi 


Mine ventilation laws, 270. 

Mines, Gases found in, 250. 

Ventilation of, 247. 

Mining,Bord-and-pillar meth¬ 
od of, 202. 

locomotive. Electric, 298. 

Longwall method of, 204. 

Machine, 246. 

Pillar-and-chamber meth¬ 
od of, 202. 

Rock-chute, 237. 

Motive column or air column, 
Calculation of, 281. 

Motor haulage, 297. 

troubles, 148. 

Motors, Alternating-current, 
149. 

Direct-current, 147. 

Efficiencies of direct-cur¬ 
rent, 162. 

Efficiencies of induction, 
163. 

Electric generators and, 
147. 

Murphy ventilator, 286. 

N 

Nasmyth fan, 284. 

Natural division of air-cur¬ 
rent, 271. 

ventilation, 279. 

New Castle, Colo., mining 
method, 227. 

Notes, Keeping surveying, 
87. 

Number of board feet in mine 
ties, Table of, 306. 

O 

Ohm’s law, 157. 

Oil for safety lamps, 255. 

Open work method of mining, 

201 . 

Opening a mine, 172. 

Orifice, The equivalent, 264. 

Overcast, 288. 

Overcome by gas, Treatment 
of persons, 318. 

P 

Pack walls, 207. 

Parallel circuits, 151. 

Percentage, 25. 


Permissible explosives, 240. 

Pillar - and - chamber method 
of mining, 202. 

-and-stall method of mi¬ 
ning, 217. 

Pillars, 209. 

Drawing, 211. 

Room, 210. 

Pins, 78. 

Piping coefficients, Table of, 
145. 

Pitch at which ant-hracite will 
run, Table of, 314. 

Pittsburg region method of 
mining, 220. 

Plane trigonometry, 58. 

Planes, Gravity, 294. 

Inclined, 95. 

Platform bars, 309. 

Plats, 193. 

Plotting, 81. 

Plumb-bob, 78. 

Position of fan, 283. 

Potential factor of a mine, 
264. 

Power calculations, 158. 
Definition of, 157. 
in alternating-current cir¬ 
cuits, 158. 

in direct - current circuits. 
158. 

of current, Horsepower or. 
260. 

or units of work per min¬ 
ute, Calculation of, 264. 
pressure and velocity, Re¬ 
lation of, 260. 
used for hoisting, 291. 

Preparation of coal, The, 307. 

Pressure and velocity, Rela¬ 
tion of power, 260. 
for box regulator. Calcu¬ 
lation of, 273. 
Measurement of, 261. 

Primary splits, 275. 

splits, Table of ventilating 
formulas for, 275. 

Problems in hoisting, 291. 

Properties of copper wire, 
161. 

of various sections, Table 

for, 101. 



xii INDEX 


Proportion, or single rule of 
three, Simple, 24. 

Proportional division of air- 
current, 272. 

Proportionate division of air, 
277. 

Props, Undersetting of, 183. 

Prospecting, 168. 

Underground, 170. 

Pulleys, 95. 

Pulsometer, 128. 

Pump machinery, 126. 

Pumps and horsepower re¬ 
quired to raise water, 
Capacity of, 127. 

Cornish, 126. 

for acid waters, 129. 

Simple and duplex, 126. 

R 

Rail elevation, 300. 

Weight of, 299. 

Rails, Table of standard size 
of, 304. 

to proper arc for any radius, 
To bend, 300. 

Rankine-Gordon formula, 27. 

Ratings, Incandescent lamp, 
163. 

Regulator, Box, 272. 

Calculation of pressure for 
box, 273. 

Door, 272. 

Side opening for door, 274. 

Relation of power, pressure, 
and velocity, 260. 

Relighting stations, 256. 

Removal of sulphur from 
coal, 313. 

Reservoirs, 125. 

Resistance, Calculation of 
mine, 263. 

Definition of, 157. 

Mine, 260. 

Revolving screen mesh for 
anthracite, 311. 

screens or trommels, 311. 

Reynoldsville region method 
of mining, 221. 

Roads and tracks, Mine, 299. 

Rock-chute mining, 237. 

Rollers, 301. 


Roof pressure, Control of, 
207. 

Room-and-pillar methods, 
Modifications of, 215. 
-and-pillar system, 202. 
pillars, 210. 

Rope calculations, Wire, 110. 
haulage, 295. 

Lay of, 104. 

Ropes, Effects of sheaves or 
drums on life of wire, 
110 . 

Flattened-strand, 104. 
Lubrication of , 112. 

Table of hoisting, haulage, 
and transmission, 106. 
Wire, 103. 

Round timber, Table for area 
of, 8. 

Rule of three, Simple propor¬ 
tion, or single, 24. 

S 

Safety catches, 293. 
lamps, 254. 
lamps, Cleaning, 256. 
lamps, Illuminating power 
of, 256. 

lamps, Locking, 256. 
lamps. Oil for, 255. 
lamps, Table of light given 
by, 257. 

lamps, Types of, 255. 

Schiele ventilator, 286. 

Speed of, 311. 

Schmidt’s law of faults, 171. 

Screen mesh for anthracite, 
Revolving, 311. 

Screens, Duty of anthracite, 
311. 

or trommels. Revolving, 
311. 

Shaking, 309. 

Table of mesh for shaking, 
310. 

Screws, 95. 

Secondary splits, 277. 

splits, Table of ventilat¬ 
ing formulas for, 276. 

Semibituminous coal, 130. 

Series circuit, 150. 

Shaft compartments, 174. 



INDEX 


xiii 


Shaft pillars, 209. 
pillars, Size of, 210. 
sinking, 174. 
timbering, 188. 

Shafts, 173. 

Finding depth of, 249. 

Size of, 174. 

Shaking screens, 309. 

screens. Table of mesh for, 
310. 

Shearing, 247. 

Sheaves, 109. 

or drums on life of wire 
ropes, Effects of, 110. 
Sheet-metal gauges, Table of 
wire and, 111. 

Shortening under stress, Elon¬ 
gation and, 102. 

Shunt, 152. 

Side opening for door reg¬ 
ulator, 274. 

Signaling, Electric, 165. 

Simple and duplex pumps, 126. 
proportion, or single rule 
of three, 24. 

Single-chute battery, 234. 
-track road. Table of mate¬ 
rial required for, 305. 
Sinking bucket, 177. 

Drainage and ventilation 
during, 178. 
engines, 178. 
head-frames, 177. 
tools, 178. 

Siphons, 120. 

Size of shaft pillars, 210. 

Sizing and classifying appa¬ 
ratus, 309. 

Slope bottoms, 302. 
sinking, 178. 

Soft, or bituminous coal, 130. 
Speed of haulage, 299. 
of screen, 311. 

Spikes required for ties, 
Table of, 305. 

Splicing a wire rope, 113. 

Splint coal, 131. 

Splits of air, Equal, 271. 
Primary, 275. 

Secondary, 277. 

Table of ventilating for¬ 
mulas for primary, 275. I 


Splits, Table of ventilating 
formulas for secondary, 
276. 

Splitting formulas, 275. 
of air-current, 270. 

Spontaneous combustion, 214. 

Square measure, 1. 
sets, 192. 

Starting strain on hoisting 
ropes, 108. 

Stations, 193. 

Relighting, 256. 

Steam, 130. 

at various pressures. Effi¬ 
ciency of, 138. 
boilers, 137. 

engine, Requirements of a 
good, 142. 
engines, 142. 

High-pressure, 138. 

Steel supports, Iron and, 195. 

Stinkdamp, 252. 

Stone dams, 124. 

Storage, Coal, 214. 

Stowings, 207. 

Strength of a beam, Break¬ 
ing, 102. 

of columns. Breaking, 102. 
of materials, 98. 

Strengths of materials, Table 
for ultimate, 100. 

Stress in hoisting ropes on in¬ 
clined planes, 105. 

Sulphur from coal. Removal 
of, 313. 

Sump, The, 179. 

Supports, Iron and steel, 195. 
Special forms of, 194. 

Surface, Metric measures of, 
4. 

Survey, Traversing a, 83. 

Surveying, 76. 
instruments, 76. 

Transit, 79. 

Underground, 85. 

Switches, 301. 

Systems of working coal, 202. 

T 

Table for bending moment 
and deflection of beams, 
100 . 




XIV 


INDEX 


Table for decimal equivalents 
of parts of 1 in., 9. 
for decimals of a foot for 
each ^-in., 10. 
for factors of safety, 99. 
for metric measures into 
English, Conversion, 7. 
for properties of various 
sections, 101. 

for stress in hoisting ropes 
on inclined planes, 105. 
Long-ton, 2. 

of composition of fuels, 136. 
of distance between cen¬ 
ters of breasts or cham¬ 
bers, 212. 

of effects of sheaves or 
drums on life of wire 
ropes, 110. 

of heating values of Amer¬ 
ican coals, 134. 
of hoisting, haulage, and 
transmission ropes, 106. 
of light given by safety 
lamp, 257. 

of loss of pressure by flow 
of air in pipes, 146. 
of materials required for 
single-track road, 305. 
of mechanical equivalents 
of electrical units, 156. 
of mesh for shaking screens, 
310. 

of mine gases, 254. 
of number of board feet in 
mine ties, 306. 
of piping coefficients, 145. 
of pitch at which anthra¬ 
cite will run, 314. 
of properties of copper 
wire, 161. 

of space occupied by 2,000 
pounds of various coals, 
315. 

of spikes required for ties, 
305. 

of standard size of rails,304. 
of ties per thousand feet 
and per mile of track, 
307. 

of trigonometric functions, 
64. 


Table of ventilation formulas, 
267. 

of ventilating formulas for 
primary splits, 275. 

of ventilating formulas for 
secondary splits, 276. 

of wire and sheet-metal 
gauges. 111. 

Tables, Metric conversion, 5. 

of logarithms, 33. 

Traverse, 12. 

Tail-rope system, 295. 

Tamping a charge, 242. 

Tape, 78. 

Telescopes, Transit, 78. 

Temperature, Measurement 
of, 263. 

Tension of haulage rope, Cal¬ 
culation of, 295. 

Tesla, Cal., mining method, 
225. 

Testing for gas by lamp flame, 
253. 

Thawing dynamite, 245. 

Thermal unit, British, 133. 

Ties, 299. 

per thousand feet and per 
mile of track, Table of, 
307. 

Timber and board measure, 8. 

and timbering, Mine, 180. 

Choice of, 180. 

head-frames or head-gears, 

200 . 

Placing of, 181. 

Table for area of cut, 9. 

Table for area of round, 8. 

Timbering a longwall face, 
207. 

and underground supports. 
Forms of mine, 183. 

Shaft, 188. 

Timbers, Preservation of, 180. 

Tools, Sinking, 178. 

Track, Gauge of, 300. 

Material required for 1,000 
ft. and for 1 M. of single, 
302. 

Table of ties per thousand 
feet and per mile of, 307. 

Tracks, Mine roads and, 299. 

1 Transformers, 150. 



INDEX 


xv 


Transit surveying, 79. 
telescopes, 78. 

The, 77. 

Transmission ropes, Table of 
hoisting, haulage, and, 
106. 

Traverse tables, 12. 

Traversing a survey, 83. 

Treatment of injured persons, 
316. 

of persons overcome by gas, 
318. 

Trestles, 195. 

Triangles, Examples in solu¬ 
tions of, 60. 

Trigonometric functions, 
Table of, 64. 

Trigonometry, Plane, 58. 

Trommels, Revolving screens 
or, 311. 

Turnouts, 302. 

Tunnels, 180. 

Types of centrifugal fans, 
284. 

U 

Ultimate strengths of mate¬ 
rials, Table for, 100. 

Undercast, 288. 

Undercutting, 202. 

Underground prospecting, 
170. 

supports, Forms of mine 
timbering and, 183. 
surveying, 85. 

Undersetting of props, 183. 

Unit, British thermal, 133. 

Units, Electrical, 155. 

Mechanical equivalents of 
electrical, 156. 
of work per minute, Calcu¬ 
lation of power or, 264. 

y 

Velocity, Measurement of, 
261. 

of air-current, 260. 

Relation of power, pres¬ 
sure, and, 260. 

Ventilating currents, Meas¬ 
urement of, 261. 
formulas for primary splits, 
Table of, 275. 


Ventilating formulas for sec¬ 
ondary splits. Table of, 
276. 

methods and appliances, 
279. 

pressure in furnace ventila¬ 
tion, Calculation of, 280. 

Ventilation, Distribution of 
air in mine, 270. 
during sinking, Drainage 
and, 178. 

Elements of, 260. 
formulas, Table of, 267. 
Furnace, 280. 
laws, Mine, 270. 

Natural, 279. 
of mines, 247. 

Quantity of air required 
for, 259. 

Ventilator, Biram, 284. 
Guibal, 286. 

Murphy, 286. 

Schiele, 286. 

Waddle, 284. 

Ventilators, Mechanical, 281. 

Volume, Metric measures of, 
4. 

W 

Waddle ventilator, 284. 

Water column corresponding 
to any mercury column, 
249. 

elevators, Miscellaneous 
forms of, 128. 

through pipes, Flow of, 119. 

Wedge, 95. 

Weight, Avoirdupois, 2. 
Metric measures of, 5. 

Weights and measures, 1. 

West Virginia method of mi¬ 
ning, 221. 

Wheel and axle, 93. 

Whitedamp, 250. 

Whiting system of hoisting, 
290. 

Wire, Aluminum, 162. 

and sheet-metal gauges. 
Table of, 111. 
data, 159. 
formulas, 162. 
gauge, 160. 

Properties of copper, 161. 






XVI 


INDEX 


Wire-rope calculations, 110. 
-rope fastenings, 113. 
rope, Splicing a, 113. 
ropes, 103. 

ropes, Effects of sheaves or 
drums on life of, 110. 
Wires, Estimation of cross- 
section of, 159. 
Estimation of resistance of, 
159. 


Wiring, Bell, 166. 

calculations for lamps, 163. 
Formulas for lamp, 163. 
Wooden dams, 124. 

Work, Definition of, 157. 
Wrought-iron chain cables, 
Resistance and proof 
tests of, 117. 



The Coal Miner’s 
Handbook 


USEFUL TABLES 


WEIGHTS AND MEASURES 


LINEAR MEASURE 


12 

inches (in.). 


= 1 foot. 


3 

feet. 


= 1 yard. 

-yd. 


yards. 


= 1 rod. 


40 

rods. 


= 1 furlong. 

. . . .fur. 

8 

furlongs. 


= 1 mile. 



in. 

ft. 

yd. rd. fur. mi. 



36 = 

3 = 

1 



198 = 

16.5 = 

5.5= 1 



7,920 = 

660 = 

220= 40=1 



63,360 = 

5,280 = 

1,760 = 320 = 8= 1 



SQUARE MEASURE 


144 square inches (sq. in.)... . 

= 1 square foot... 


9 square feet. 

= 1 square yard.. 


30 J square yards. 

= 1 square rod... 


160 square rods. 

= 1 acre. 

. .....A. 

640 acres. 

= 1 square mile... 


sq. mi. A. sq. rd. sq. yd. 

sq. ft. 

sq. in. 


1 = 640 = 102,400 = 3,097,600 = 27,878,400 = 4,014,489,600 

























2 


USEFUL TABLES 


MEASURE OF ANGLES OR ARCS 

60 seconds (").=1 minute. 

60 minutes.=1 degree.° 

90 degrees.= 1 rt. angle or quadrant. . . . □ 

360 degrees.= 1 circle.cir. 

1 cir. = 360° = 21,600'= 1,296,000" 


CUBIC MEASURE 

1,728 cubic inches (cu. in.). . .. = 1 cubic foot. 


. cu, 


ft. 


27 cubic feet. 
128 cubic feet. 


= 1 cubic yard.cu. yd. 

= 1 cord.cd. 


24 f cubic feet.=1 perch. 


,P. 


cu. yd. 

1 = 


cu. ft. cu. in. 
27 = 46,656 


AVOIRDUPOIS WEIGHT 


437| grains (gr.). 

16 ounces. 

100 pounds. 

20 cwt., or 2,000 lb 


= 1 ounce. 

= 1 pound. 

= 1 hundredweight 
= 1 ton. 


T. cwt. lb. oz. gr. 

1 = 20 = 2,000 = 32,000 = 14,000,000 
The avoirdupois pound contains 7,000 gr. 


.oz. 
. .lb. 
cwt. 
. .T. 


LONG-TON TABLE 


16 ounces. 

112 pounds. 

20 cwt., or 2,240 lb.. 


.lb. 

. . . .cwt. 
.T. 

2 pints (pt.). 

8 quarts. 

4 pecks. 

DRY MEASURE 

bu. pk. qt. pt. 

1 = 4 = 32 = 64 

.qt. 

.pk. 

.bu. 


The U. S. struck bushel contains 2,150.42 cu. in. = 1.2444 
cu. ft. By law, its dimensions are those of a cylinder 18! in. 
in diameter and 8 in. deep. The heaped bushel is equal to 
1! struck bu., the cone being 6 in. high. For approximations, 




































USEFUL TABLES 3 

the bushel may be taken at If cu. ft. or 1 cu. ft. may be con¬ 
sidered I bu. 

The British bushel contains 2,218.19 cu. in. = 1.2837 cu. ft. 
= 1.032 U. S. bu. 

The dry gallon contains 268.8 cu. in., or f struck bu. 


LIQUID MEASURE 


4 gills (gi.). 


.Pt. 

2 pints. 


.qt. 

4 quarts. 

. . = 1 gallon. 


31 § gallons. 


.bbl. 

2 barrels, or 63 gallons.... 


.hhd. 


hhd. bbl. gal. qt. pt. gi. 

1 = 2 = 63 = 252 = 504 = 2,016 
The U. S. gallon contains 231 cu. in. = .134 cu. ft., nearly; 
or 1 cu. ft. contains 7.481 gal. The following cylinders con¬ 
tain the given measures very closely: 

Diam. Height Diam. Height 

Inches Inches Inches Inches 


Gill. 

.If 

3 

Gallon. 

. 7 

6 

Pint. 

.3| 

3 

8 gal. 

.14 

12 

Quart. 

.3| 

6 

10 gal. 

.14 

15 


When water is at its maximum density, 1 cu. ft. weighs 
62.425 lb. and 1 gal. weighs 8.345 lb. 


For approximations, 1 cu. ft. of water is considered equal 
to 7 5 gal., and 1 gal. as weighing 8 f lb. 

The British imperial gallon, both liquid and dry, contains 
277.274 cu. in. = .16046 cu. ft., and is equivalent to the volume 
of 10 lb. of pure water at 62° F. 

To reduce British to U. S. liquid gallons, multiply by 1.2. 
Conversely, to convert U. S. into British liquid gallons, divide 
by 1 . 2 ; or, increase the number of gallons one-fifth. 


METRIC SYSTEM 

The metric system is based on the meter, which, according to 
the U. S. Coast and Geodetic Survey report of 1884, is equal 
to 39.370432 in. The value commonly used is 39.37 in., and 
is authorized by the U. S. government. The meter is defined 






















4 


USEFUL TABLES 


as one ten-millionth the distance from the pole to the equator 
measured on a meridian passing near Paris. 

There are three principal units—the meter, the liter (pro¬ 
nounced lee-ter), and the gram, the units of length, capacity, 
and weight, respectively. Multiples of these units are obtained 
by prefixing to the names of the principal units the Greek 
words deca ( 10 ), hecto ( 100 ), and kilo ( 1 , 000 ); the submulti¬ 
ples, or divisions, are obtained by prefixing the Latin words 
deci (in), ccnti (xihx), and milli (toW). These prefixes form 
the key to the entire system. In the following tables, the 
abbreviations of the principal units of these submultiples begin 
with a small letter, while those of the multiples begin with a 
capital letter; they should always be written as here printed. 


MEASURE 

OF LENGTH 


10 millimeters (mm.) . 

= 1 centimeter . 


10 centimeters . 

= 1 decimeter . 


10 decimeters . 

= 1 meter . 


10 meters . 

= 1 decameter . 


10 decameters . 

= 1 hectometer . 

.... Hm. 

10 hectometers . 

= 1 kilometer . 

.... Km. 

MEASURES OF SURFACE (NOT LAND) 


100 square millimeters(sq.mm.) 

= 1 square centimeter. 

. . sq. cm. 

100 square centimeters . 

= 1 square decimeter.. 

. .sq. dm. 

100 square decimeters . 

= 1 square meter . 

. . .sq. m. 

MEASURES 

OF VOLUME 


1,000 cubic millimeters . 

= 1 cubic centimeter 


(cu. mm.) 

. c. c. or cu. cm. 

1,000 cubic centimeters . 

= 1 cubic decimeter. .. 


1,000 cubic decimeters . 

= 1 cubic meter . 

.. .cu. m. 

MEASURES 

OF CAPACITY 


10 milliliters (ml.) . 

— 1 centiliter . 


10 centiliters . 

= 1 deciliter . 

. dl. 

10 deciliters . 

= 1 liter . 

. 1 . 

10 liters. 

. = 1 decaliter. 

.Dl. 

10 decaliters. 

. = 1 hecoliters. 

.HI. 

10 hecoliters . 

. = 1 kiloliters . 

. Kl. 

The liter is equal to the volume occupied by 1 cu. 

dm. 


































USEFUL TABLES 


5 


MEASURES OF WEIGHT 

10 milligrams (mg.).=1 centigram.eg. 

10 centigrams.=1 decigram.dg. 

10 decigrams. 

10 grams. 

10 decagrams. 

10 hectograms. 

1,000 kilograms. 


.=1 gram.g. 

.=1 decagram.Dg. 

.=1 hectogram.Hg. 

.= 1 kilogram.Kg. 

.=1 ton.T. 

„ The gram is the weight of 1 c. c. of pure distilled water at 
a temperature of 39.2° F.; the kilogram is the weight of 1 1. 
of water; the ton is the weight of 1 cu. m. of water. 


1,828.8 

121.92 

21.336 

.3048 

.2438 

1,972.6046 


CONVERSION TABLES 

By means of the accompanying tables, metric measures 
can be converted into English and vice versa, by simple addi¬ 
tion. All the figures of the values given are not required 
except in very exact calculations; as a rule, 4 or 5 digits only 
are used. To change 6,471.8 ft. into meters, con¬ 
sider 6,471.8 as 6.000+400 + 70 + 1 + .8; also, 6,000 
= 1,000X6; 400=100X4, etc. Hence, looking in 
the first column of the table entitled English Meas¬ 
ures Into Metric, for 6 (the first figure of the given 
number), opposite it in the column headed Feet to 
Meters, is found the number 1.8287838. Using but 
five digits and increasing the fifth digit by 1 (as 
the next is greater than 5), gives 1.8288. In other words, 
6 ft. = 1.8288 m.; hence, 6,000 ft. = 1,000X 1.8288= 1,828.8, 
simply moving the decimal point three places to the right. 
Likewise, it is found that 400 ft. = 121.92 m.; 70 ft. = 21.336 m.; 
1 ft. = .3048 m., and .8 ft. = .2438 m. Adding as 
shown, gives 1,972.6046 m.as the value of 6,471.8 ft. 22.046 
As another example, convert 19.635 kg. into 19.8416 

pounds. Working according to the explanation 1.3228 

just given, it is found that 19.635 kg. = 43.2875 lb. .0661 

The only difficulty in applying these tables lies .0110 

in locating the decimal point; it may always be - 

found thus: If the figure considered lies to the left 43.2875 
of the decimal point, count each figure in order, 
beginning with units (but calling units’ place zero), until the 




















6 


USEFUL TABLES 


desired figure is reached, then move the decimal point to the 
right as many places as the figure being considered is to the 


CONVERSION TABLE 

English Measures Into Metric 


Eng¬ 

lish 

Metric 

Metric 

Metric 

Metric 

Inches to 

Feet to 

Pounds to 

Gallons to 


Meters 

Meters 

Kilos 

Liters 

1 

.0253998 

.3047973 

.4535925 

3.7853122 

2 

.0507996 

.6095946 

.9071850 

7.5706244 

3 

.0761993 

1 .9143919 

1.3607775 

11.3559366 

4 

.1015991 

1.2191892 

1.8143700 

15.1412488 

5 

.1269989 

1.5239865 

2.2679625 

18.9265610 

6 

.1523987 

1.8287838 

2.7215550 

22.7118732 

7 

.1777984 

2.1335811 

3.1751475 

26.4971854 

8 

.2031982 

2.4383784 

3.6287400 

30.2824976 

9 

.2285980 

2.7431757 

4.0823325 

34.0678098 

10 

.2539978 

3.0479730 

4.5359250 

37.8531220 


Metric 

Metric 

Metric 

Metric 


Square 

Square 

Cubic 

Pounds per 

Eng¬ 

lish 

Inches 

Feet 

Feet 

Square Inch 

to 

to 

to 

to Kilo per 

Square 

Square 

Cubic 

Square 


Meters 

Meters 

Meters 

Meter 

1 

.000645150 

.092901394 

.028316094 

703.08241 

2 

.001290300 

.185802788 

.056632188 

1,406.16482 

3 

.001935450 

.278704182 

.084948282 

2,109.24723 

4 

.002580600 

.371605576 

.113264376 

2,812.32964 

5 

.003225750 

.464506970 

.141580470 

3,515.41205 

6 

.003870900 

.557408364 

.169896564 

4,218.49446 

7 

.004516050 

.650309758 

.198212658 

4,921.57687 

8 

.005161200 

.743211152 

.226528752 

5,624.65928 

9 

.005806350 

.836112546 

.254844846 

6,327.74169 

10 

.006451500 

.929013940 

.283160940 

7,030.82410 


left of the unit figure. Thus, in the first example, 6 lies three 
places to the left of 1, which is in units’ place; hence, the deci¬ 
mal point is moved three places to the right. By exchanging 

































USEFUL TABLES 


7 


the words right and left, the statement will also apply to deci¬ 
mals. Thus, in the second case above, the 5 lies three places 


CONVERSION TABLE 

Metric Measures Into English 


Metric 

English 

English 

English 

English 

Meters to 

Meters to 

Kilos to 

Liters to 



Inches 

Feet 

Pounds 

Gallons 

1 

39.370432 

3.2808693 

2.2046223 

.2641790 

2 

78.740864 

6.5617386 

4.4092447 

.5283580 

3 

118.111296 

9.8426079 

6.6138670 

.7925371 

4 

157.481728 

13.1234772 

8.8184894 

1.0567161 

5 

196.852160 

16.4043465 

11.0231117 

1.3208951 

6 

236.222592 

19.6852158 

13.2277340 

1.5850741 

7 

275.593024 

22.9660851 

15.4323564 

1.8492531 

8 

314.963456 

26.2469544 

17.6369787 

2.1134322 

9 

354.333888 

29.5278237 

19.8416011 

2.3776112 

10 

393.704320 

32.8086930 

22.0462234 

2.6417902 


English 

English 

English 

English 


Square 

Square 

Cubic 

Kilos per 
Square 
Meter to 
Pounds per 
Square 
Inch 

Metric 

Meters 

to 

Square 

Meters 

to 

Square 

Meters 

to 

Cubic 


Inches 

Feet 

Feet 

1 

1,550.03092 

10.7641034 

35.3156163 

.001422310 

2 

3,100.06184 

21.5282068 

70.6312326 

.002844620 

3 

4,650.09276 

32.2923102 

105.9468489 

.004266930 

4 

6,200.12368 

43.0564136 

141.2624652 

.005689240 

5 

7,750.15460 

53.8205170 

176.5780815 

.007111550 

6 

9,300.18552 

64.5846204 

211.8936978 

.008533860 

7 

10,850.21644 

75.3487238 

247.2093141 

.009956170 

8 

12,400.24736 

86.1128272 

282.5249304 

.011378480 

9 

13,950.27828 

96.8769306 

317.8405467 

.012800790 

10 

15,500.30920 

107.6410340 

353.1561630 

.014223100 


to the right of units’ place; hence, the decimal point in the 
number taken from the table is moved three places to the left. 






































8 


USEFUL TABLES 


TIMBER AND BOARD MEASURE 

TIMBER MEASURE 

Volume of Round Timber.—The volume of round timber, in 
cubic feet, equals the length multiplied by one-fourth the prod¬ 
uct of mean girth and diameter, all dimensions being in feet. 
If length is given in feet and girth and diameter in inches, 
divide by 144; if all dimensions are in inches, divide by 1,728. 


AREA OF ROUND TIMBER 


1 Girths 
Inches 

Area 

Square 

Feet 

1 Girths 
Inches 

Area 

Square 

Feet 

1 Girths 
Inches 

Area 

Square 

Feet 

6 

.250 

121 

1.04 

19 

2.50 

6i 

.272 

121 

1.08 

19* 

2.64 

6* 

.294 

12f 

1.12 

20 

2.77 

6f 

.317 

13 

1.17 

20* 

2.91 

7 

.340 

131 

1.21 

21 

3.06 

7 i 

.364 

131 

1.26 

21* 

3.20 

7 * 

.390 

13f 

1.31 

22 

3.36 

7f 

.417 

14 

1.36 

22* 

3.51 

8 

.444 

141 

1.41 

23 

3.67 

81 

.472 

141 

1.46 

23* 

3.83 

8* 

.501 

14f 

1.51 

24 

4.00 

8f 

.531 

15 

1.56 

24* 

4.16 

9 

.562 

151 

1.61 

25 

4.34 

91 

.594 

151 

1.66 

25* 

4.51 

9* 

.626 

151 

1.72 

26 

4.69 

9f 

.659 

16 

1.77 

26* 

4.87 

10 

.694 

161 

1.83 

27 

5.06 

101 

.730 

161 

1.89 

27* 

5.25 

101 

.766 

161 

1.94 

28 

5.44 

101 

.803 

17 

2.00 

28* 

5.64 

11 

.840 

171 

2.09 

29 

5.84 

m 

.878 

171 

2.12 

29* 

6.04 

111 

.918 

171 

2.18 

30 

6.25 

Ilf 

.959 

18 

2.25 



12 

1.000 

181 

2.37 




In the accompanying table is given the area of round timber. 
The area corresponding to 1 girth (mean), in inches, multiplied 
by the length, in feet, equals the solidity, in feet and decimal 
parts. 
















USEFUL TABLES 


9 


AREA OF CUT TIMBER 


Breadth 

Inches 

Area of 

1 Lin. Ft. 

Breadth 

Inches 

Area of 

1 Lin. Ft. 

Breadth 

Inches 

Area of 

1 Lin. Ft. 

1 

.021 

41 

.354 

81 

.688 

h 

.042 

41 

.375 

81 

.708 

3 

4 

.063 

41 

.396 

81 

.729 

1 

.083 

5 

.417 

9 

.750 

U 

.104 

51 

.438 

91 

.771 

H 

.125 

51 

.458 

91 

.792 

if 

.146 

51 

.479 

91 

.813 

2 

.167 

6 

.500 

10 

.833 

21 

.188 

61 

.521 

101 

.854 

2i 

.208 

61 

.542 

101 

.875 

2f 

.229 

61 

.563 

101 

.896 

3 

.250 

7 

.583 

11 

.917 

31 

.271 

71 

.604 

HI 

.938 

31 

.292 

71 

.625 

111 

.958 

31 

.313 

71 

.646 

111 

.979 

4 

.333 

8 

.667 

12 

1.000 


DECIMAL EQUIVALENTS OF PARTS OF 1 IN. 



4-3 


4-3 


43> 


4-3 


G 




C 


c 

° ^ 

CD 

U-4 

0 

CD 

<4-J 

° A 

0) 

9-1 

0 rC 

(D 

4-J o 
05 £ 

P< 

> 

• H 

P 

cr 

4-3 o 

05 ^ 

Ph 

'3 

o' 

O 

0j 

P-l 

> 

'3 

c 

Ph 

> 

'3 

o' 


w 


W 


w 


W 

57 

.015625 

17 

64 

.265625 

33 

64 

.515625 

49 

64 

.765625 

T? 

.031250 

9 

32 

.281250 

17 

32 

.531250 

25 

32 

.781250 

A 

.046875 

19 

64 

.296875 

35 

64 

.546875 

51 

64 

.796875 

TS 

.062500 

5 

1 6 

.312500 

9 

16 

.562500 

1 3 

16 

.812500 

<rr 

.078125 

21 

FI 

.328125 

37 

64 

.578125 

53. 

64 

.828125 

T2 

.093750 

11 

32 

.343750 

19 

32 

.593750 

27 

32 

.843750 

7 

64 

.109375 

2_3 

64 

.359375 

39 

64 

.609375 

55 

64 

.859375 

1 

8 

.125000 

3 

8 

.375000 

5 

8 

.625000 

7 

8 

.875000 


.140625 

25 

64 

.390625 

41 

64 

.640625 

57 

64 

.890625 


.156250 

13 

32 

.406250 

2_1_ 

32 

.656250 

29 

32 

.906250 


.171875 

27 

64 

.421875 

43 

64 

.671875 

59 

64 

.921875 

T6 

.187500 

7 

16 

.437500 

11 

16 

.687500 

15 

1 6 

.937500 

H 

.203125 

29 

64 

.453125 

45 

64 

,703125 

.61 

64 

.953125 

T2 

.218750 

15 

32 

.468750 

23 

32 

.718750 

ft 

.968750 

ft 

.234375 

31 

64 

.484375 

47 

64 

.734375 

63. 

64 

.984375 

1 

4 

.250000 

1 

2 

.500000 

3 

4 

.750000 

1 

1 






































10 


USEFUL TABLES 


DECIMALS OF A FOOT FOR EACH 1-32 IN. 


In. 

0" 

1" 

2" 

3" 

4" 

5" 

0 

0 

.0833 

.1667 

.2500 

.3333 

.4167 

"hi 

.0026 

.0859 

.1693 

.2526 

.3359 

.4193 

it 

.0052 

.0885 

.1719 

.2552 

.3385 

.4219 

T2 

.0078 

.0911 

.1745 

.2578 

.3411 

.4245 

1 

.0104 

.0937 

.1771 

.2604 

.3437 

.4271 

_5_ 

.0130 

.0964 

.1797 

.2630 

.3464 

.4297 


.0156 

.0990 

.1823 

.2656 

.3490 

.4323 

7 

.0182 

.1016 

.1849 

.2682 

.3516 

.4349 

1 

.0208 

.1042 

.1875 

.2708 

.3542 

.4375 

9 

32 

.0234 

.1068 

.1901 

.2734 

.3568 

.4401 

5 

16 

ft 

.0260 

.1094 

.1927 

.2760 

.3594 

.4427 

.0286 

.1120 

.1953 

.2786 

.3620 

.4453 

3 

8 

.0312 

.1146 

.1979 

.2812 

.3646 

.4479 

13 

32 

.0339 

.1172 

.2005 

.2839 

.3672 

.4505 

7 

16 

.0365 

.1198 

.2031 

.2865 

.3698 

.4531 

15 

32 

.0391 

.1224 

.2057 

.2891 

.3724 

.4557 

1 

2 

.0417 

.1250 

.2083 

.2917 

.3750 

.4583 

ft 

.0443 

.1276 

.2109 

.2943 

.3776 

.4609 

TW 

.0469 

.1302 

.2135 

.2969 

.3802 

.4635 

19 

32 

.0495 

.1328 

.2161 

.2995 

.3828 

.4661 

5 

8 

.0521 

.1354 

.2188 

.3021 

.3854 

.4688 

21 

32 

.0547 

.1380 

.2214 

.3047 

.3880 

.4714 

fs 

.0573 

.1406 

.2240 

.3073 

.3906 

.4740 

23 

32 

.0599 

.1432 

.2266 

.3099 

.3932 

.4766 

3 

4 

.0625 

.1458 

.2292 

.3125 

.3958 

.4792 

ft 

.0651 

.1484 

.2318 

.3151 

.3984 

.4818 

JL3 

16 

.0677 

.1510 

.2344 

.3177 

.4010 

.4844 

2 7 

32 

.0703 

.1536 

.2370 

.3203 

.4036 

.4870 

7 

8 

.0729 

.1562 

.2396 

.3229 

.4062 

.4896 

ft 

.0755 

.1589 

.2422 

.3255 

.4089 

.4922 

15 

16 

.0781 

.1615 

.2448 

.3281 

.4115 

.4948 

ft 

.0807 

.1641 

.2474 

.3307 

.4141 

.4974 


Volume of Square Timber.—When all dimensions are in feet: 
Rule. —Multiply the breadth by the depth and that product by 
the length, and the product will give the volume, in cubic feet. 
When either of the dimensions is in inches: 

Rule. —Multiply as before and divide by 12. 

When any two of the dimensions are in inches: 

Rule. —Multiply as before and divide by IJ+J+. 















USEFUL TABLES 


11 


Table —( Continued) 


In. 

6" 

7" 

8" 

9" 

10" 

11" 

0 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 


.5026 

.5859 

.6693 

.7526 

.8359 

.9193 

TS 

.5052 

.5885 

.6719 

.7552 

.8385 

.9219 

Y2 

.5078 

.5911 

.6745 

.7578 

.8411 

.9245 

1 

.5104 

.5937 

.6771 

.7604 

.8437 

.9271 


.5130 

.5964 

.6797 

.7630 

.8464 

.9297 

A 

.5156 

.5990 

.6823 

.7656 

.8490 

.9323 

T2 

.5182 

.6016 

.6849 

.7682 

.8516 

.9349 

I 

4 

.5208 

.6042 

.6875 

.7708 

.8542 

.9375 

T2 

.5234 

.6068 

.6901 

.7734 

.8568 

.9401 

A 

H 

.5260 

.6094 

.6927 

.7760 

.8594 

.9427 

.5286 

.6120 

.6953 

.7786 

.8620 

.9453 

3 

8 

.5312 

.6146 

.6979 

.7812 

.8646 

.9479 

H 

.5339 

.6172 

.7005 

.7839 

.8672 

.9505 

IdS 

.5365 

.6198 

.7031 

.7865 

.8698 

.9531 


.5391 

.6224 

.7057 

.7891 

.8724 

.9557 

* 

.5417 

.6250 

.7083 

.7917 

.8750 

.9583 

a 

.5443 

.6276 

.7109 

.7943 

.8776 

.9609 

A 

.5469 

.6302 

.7135 

.7969 

.8802 

.9635 

19 

32 

.5495 

.6328 

.7161 

.7995 

.8828 

.9661 

5 

8 

.5521 

.6354 

.7188 

.8021 

.8854 

.9688 

21 

32 

.5547 

.6380 

.7214 

.8047 

.8880 

.9714 

UL 

16 

.5573 

.6406 

.7240 

.8073 

.8906 

.9740 

23 

3~2 

.5599 

.6432 

.7266 

.8099 

.8932 

.9766 

3 

.5625 

.6458 

.7292 

.8125 

.8958 

.9792 

25 

32 

.5651 

.6484 

.7318 

.8151 

.8984 

.9818 

13 

.5677 

.6510 

.7344 

.8177 

.9010 

.9844 

27 

.5703 

.6536 

.7370 

.8203 

.9036 

.9870 

7 

8 

.5729 

.6562 

.7396 

.8229 

.9062 

.9896 

M 

.5755 

.6589 

.7422 

.8255 

.9089 

.9922 

15 

.5781 

.6615 

.7448 

.8281 

.9115 

.9948 

3J, 

32 

.5807 

.6641 

.7474 

.8307 

.9141 

.9974 


BOARD MEASURE 

In measuring, boards are assumed to be 1 in. thick. The 
number of feet, board measure (B. M.), in a given board or 
stick of timber, equals the length, in feet, multiplied by the 
breadth, in feet, multiplied by the thickness, in inches. 

Area of 1 lin. ft. multiplied by length, in feet, will give super¬ 
ficial contents, in square feet. 






















12 


USEFUL TABLES 


DECIMAL EQUIVALENTS 

In many cases of taking measurements, it is desirable to 
change a fraction of an inch or foot to decimals, or getting the 
nearest fraction of an inch or foot from a calculation in 
which a large decimal appears. The preceding tables give the 
decimal equivalents of each b 1 ? in. and the decimal equivalents 
of 1 ft. for each ft in. 


TRAVERSE TABLES 

To use the traverse tables, find the number of degrees in the 
left-hand column if the angle is less than 45°, and in the right- 
hand column if greater than 45°. The numbers on the same 
line running across the page are the latitudes and departures 
for that angle and for the respective distances, 1, 2, 3, 4, 5, 
6, 7, 8, 9, which appear at the top of the pages. Thus, if the 
bearing of a line is 10° and the distance is 4, the latitude will be 
3.939 and the departure .695; with the same bearing, and the 
distance 8, the latitude will be 7.878 and the departure 1.389. 
The latitude and departure for 80 is 10 times the latitude 
and departure for 8, and is found by moving the decimal point 
one place to the right; that for 500 is 100 times the latitude and 
departure for 5, and is ound by moving the decimal point two 
places to the right and so on. By moving the decimal point 
one, two, or more places to the right, the latitude and departure 
may be found for any multiple of any number given in the 
table. In finding the latitude and departure for any number 
such as 453, the number is resolv ed into three numbers, viz. 
400, 50, 3, and the latitude and departure for each is taken 
from the table and then added together. 

Rule .—Write down the latitude and departure, neglecting the 
decimal points, for the first figure of the given distance; write 
under them the latitude and departure for the second figure, setting 
them one place farther to the right; under these, place the latitude 
and departure for the third figure, setting them one place still 
farther to the right, and so continue until all the figures of the given 
distance have been used; add these latitudes and departures, and 
Point off on the right of their sums a number of decimal places 



USEFUL TABLES 


13 


equal to the number of decimal places to which the tables being 
used are carried; the resulting numbers will be the latitude and 
departure of the given distance in feet, links, chains, or whatever 
unit of measurement is adopted. Should the departure or latitude 
consist only of a decimal, a cipher should be inserted before the 
decimal, as in the departures of example 1. 

Example 1 . —A bearing is 16° and the distance 725 ft.; 
what is the latitude and departure? 

Solution. —Applying the rule just given: 


Distances 
700 
2 0 
5 


Latitudes 
6 729 
19 2 3 
4 8 0 6 


Departures 
19 2 9 
0 5 5 1 
13 78 


7 2 5 6 9 6.9 3 6 1 9 9.7 8 8 


Taking the nearest whole numbers and rejecting the decimals, 
the latitude and departure are 697 and 200. 

When a 0 occurs in the given number, the next figure must 
be set two places to the right as in the following example: 

Example 2.—The bearing is 22° and the distance 907 ft.; 
required, the latitude and departure. 

Solution. —Applying the rule just given: 


Distances 

900 

7 

9 0 7 


Latitudes 
8 3 4 5 
6 4 9 0 
8 4 0.9 9 0 


Departures 
33 7 1 
2 6 2 2 
3 3 9.7 2 2 


Here the place of 0 both in the distance column and in the 
latitude and departure columns is occupied by a dash —. 
Rejecting the decimals, the latitude is 841 ft. and the depart¬ 
ure 340 ft. 

When the bearing is more than 45°, the names of the columns 
must be read from the bottom of the page. The latitude of 
any bearing, as 60°, is the departure of its complement, 30°; 
and the departure of any bearing, as 30°, is the latitude of its 
complement, 60°. Where the bearings are given in smaller 
fractions of degrees than is found in the table, the latitudes 
and departures can be found by interpolation. 










14 


USEFUL TABLES 


bo 

B <U 
C ^ 

1 

2 

3 

A 


5 

a 










u H 

(1) 










c* $ 

d; 

35 Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

CQ Q 

0 

1.000 

.000 

2.000 

.000 

3.000 

.000 

4.000 

.000 

5.000 

90 

1 

1.000 

.004 

2.000 

.009 

3.000 

.013 

4.000 

.017 

5.000 

891 

1 

1.000 

.009 

2.000 

.017 

3.000 

.026 

4.000 

.035 

5.000 

89 1 

3 

4 

1.000 

.013 

2.000 

.026 

3.000 

.039 

4.000 

.052 

5.000 

891 

1 

1.000 

.017 

2.000 

.035 

3.000 

.052 

3.999 

.070 

4.999 

89 

a 

1.000 

.022 

2.000 

.044 

2.999 

.065 

3.999 

.087 

4.999 

881 

ll 

1.000 

.026 

1.999 

.052 

2.999 

.079 

3.999 

.105 

4.998 

881 

11 

1.000 

.031 

1.999 

.061 

2.999 

.092 

3.998 

.122 

4.998 

881 

2 

.999 

.035 

1.999 

.070 

2.998 

.105 

3.998 

.140 

4.997 

88 

2j 

.999 

.039 

1.998 

.079 

2.998 

.118 

3.997 

.157 

4.996 

871 

21 

.999 

.044 

1.998 

.087 

2.997 

.131 

3.996 

.174 

4.995 

871 

2| 

.999 

.048 

1.998 

.096 

2.997 

.144 

3.995 

.192 

4.994 

871 

3 

.999 

.052 

1.997 

.105 

2.996 

.157 

3.995 

.209 

4.993 

87 

3? 

.998 

.057 

1.997 

.113 

2.995 

.170 

3.994 

.227 

4.992 

861 

31 

.998 

.061 

1.996 

.122 

2.994 

.183 

3.993 

.244 

4.991 

861 

3! 

.998 

.065 

1.996 

.131 

2.994 

.196 

3.991 

.262 

4.989 

861 

4 

.998 

.070 

1.995 

.140 

2.993 

.209 

3.990 

.279 

4.988 

86 

41 

.997 

.074 

1.995 

.148 

2.992 

.222 

3.989 

.296 

4.986 

851 

41 

.997 

.078 

1.994 

.157 

2.991 

.235 

3.988 

.314 

4.985 

851 

41 

.997 

.083 

1.993 

.166 

2.990 

.248 

3.986 

.331 

4.983 

851 

5 

.996 

.087 

1.992 

.174 

2.989 

.261 

3.985 

.349 

4.981 

85 

51 

.996 

.092 

1.992 

.183 

2.987 

.275 

3.983 

.366 

4.979 

841 

51 

.995 

.096 

1.991 

.192 

2.986 

.288 

3.982 

.383 

4.977 

841 

51 

.995 

.100 

1.990 

.200 

2.985 

.301 

3.980 

.401 

4.975 

841 

6 

.995 

.105 

1.989 

.209 

2.984 

.314 

3.978 

.418 

4.973 

S4 

61 

.994 

.109 

1.988 

.218 

2.9S2 

.327 

3.976 

.435 

4.970 

831 

61 

.994 

.113 

1.987 

.226 

2.981 

.340 

3.974 

.453 

4.968 

831 

61 

.993 

.118 

1.986 

.235 

2.979 

.353 

3.972 

.470 

4.965 

831 

7 

.993 

.122 

1.985 

.244 

2.978 

.366 

3.970 

.487 

4.963 

83 

71 

.992 

.126 

1.984 

.252 

2.978 

.379 

3.968 

.505 

4.960 

821 

71 

.991 

.131 

1.9S3 

.261 

2.974 

.392 

3.966 

.522 

4.957 

821 

72 

.991 

.135 

1.982 

.270 

2.973 

.405 

3.963 

.539 

4.954 

821 

8 

.990 

.139 

1.981 

.278 

2.971 

.418 

3.961 

.557 

4.951 

82 

81 

.990 

.143 

1.979 

.287 

2.969 

.430 

3.959 

.574 

4.948 

811 

81 

.989 

.148 

1.978 

.296 

2.967 

.443 

3.956 

.591 

4.945 

811 

81 

.988 

.152 

1.977 

.304 

2.965 

.456 

3.953 

.608 

4.942 

811 

9 

.988 

.156 

1.975 

.313 

2.963 

.469 

3.951 

.626 

4.938 

81 

bo w 

.5 ^ 
C *■« 

aj bO 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

bo £ 

2 <D 










0 ? bo 

eq Q 

1 


> 

3 

4 

5 

m Q 






































USEFUL TABLES 15 


be m 
c ^ 
~ 0) 

5 

6 

7 

8 


3 

bo £ 












Ci SP 










aj bO 

pq Q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

m Q 

0 

.000 

6.000 

.000 

7.000 

.000 

8.000 

.000 

9.000 

.000 

90 

1 

4 

.022 

6.000 

.026 

7.000 

.031 

8.000 

.035 

9.000 

.039 

891 

1 

2 

.044 

6.000 

.052 

7.000 

.061 

8.000 

.070 

9.000 

.079 

89 4 

3 

4 

.065 

5.999 

.079 

6.999 

.092 

7.999 

.105 

8.999 

.118 

89 i 

1 

.087 

5.999 

.105 

6.999 

.122 

7.999 

.140 

8.999 

.157 

89 

U 

.109 

5.999 

.131 

6.998 

.153 

7.998 

.175 

8.998 

.196 

88f 

14 

.131 

5.998 

.157 

6.998 

.183 

7.997 

.209 

8.997 

.236 

881 

U 

.153 

5.997 

.183 

6.997 

.214 

7.996 

.244 

8.996 

.275 

88j 

2 

.174 

5.996 

.209 

6.996 

.244 

7.995 

.279 

8.995 

.314 

88 

2i 

.196 

5.995 

.236 

6.995 

.275 

7.994 

.314 

8.993 

.353 

871 

2? 

.218 

5.994 

.262 

6.993 

.305 

7.992 

.349 

8.991 

.393 

87 5 

2f 

.240 

5.993 

.288 

6.992 

.336 

7.991 

.384 

8.990 

.432 

87 J 

3 

.262 

5.992 

.314 

6.990 

.366 

7.989 

.419 

8.988 

.471 

87 

3 ? 

.283 

5.990 

.340 

6.989 

.397 

7.987 

.454 

8.986 

.510 

86 f 

34 

.305 

5.989 

.366 

6.987 

.427 

7.985 

.488 

8.983 

.549 

86 4 

3f 

.327 

5.987 

.392 

6.985 

.458 

7.983 

.523 

8.981 

.589 

86 J 

4 

.349 

5.985 

.419 

6.983 

.488 

7.981 

.558 

8.978 

.628 

86 

4f 

.371 

5.984 

.445 

6.981 

.519 

7.978 

.593 

S.975 

.667 

85 f 

4? 

.392 

5.982 

.471 

6.978 

.549 

7.975 

.628 

8.972 

.706 

854 

4f 

.414 

5.979 

.497 

6.976 

.580 

7.973 

.662 

8.969 

.745 

851 

5 

.436 

5.977 

.523 

6.973 

.610 

7.970 

.697 

8.966 

.784 

85 

51 

.458 

5.975 

.549 

6.971 

.641 

7.966 

.732 

8.962 

.824 

84 f 

51 

.479 

5.972 

.575 

6.968 

.671 

7.963 

.767 

8.959 

.863 

841 

5f 

.501 

5.970 

.601 

6.965 

.701 

7.960 

.802 

8.955 

.902 

84 J 

6 

.523 

5.967 

.627 

6.962 

.732 

7.956 

.836 

8.951 

.941 

84 

61 

.544 

5.964 

.653 

6.958 

.762 

7.952 

.871 

8.947 

.980 

83 f 

61 

.566 

5.961 

.679 

6.955 

.792 

7.949 

.906 

8.942 

1.019 

83 * 

6! 

.58S 

5.958 

.705 

6.951 

.823 

7.945 

.940 

8.938 

1.058 

831 

7 

.609 

5.955 

.731 

6.948 

.853 

7.940 

.975 

8.933 

1.097 

83 

71 

.631 

5.952 

.757 

6.944 

.883 

7.936 

1.010 

8.928 

1.136 

821 

71 

.653 

5.949 

.783 

6.940 

.914 

7.932 

1.044 

8.923 

1.175 

82 5 

7! 

.674 

5.945 

.809 

6.936 

.944 

7.927 

1.079 

8.918 

1.214 

824 

8 

.696 

5.942 

.835 

6.932 

.974 

7.922 

1.113 

8.912 

1.253 

82 

81 

.717 

5.938 

.861 

6.928 

1.004 

7.917 

1.148 

8.907 

1.291 

81J 

81 

.739 

5.934 

.8S7 

6.923 

1.035 

7.912 

1.182 

8.901 

1.330 

81| 

Sf 

.761 

5.930 

.913 

6.919 

1.065 

7.907 

1.217 

8.895 

1.369 

814 

9 

.782 

5.926 

.939 

6.914 

1.095 

7.902 

1.251 

8.889 

1.408 

81 

bo s 
.5 g 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

tso “ 

r*l D 

.5 a) 

I- h 

rt M 











<L> <13 

pq Q 

5 

€ 


7 


S 


9 

pq Q 












































16 USEFUL TABLES 


to to 

r> <U 

.5 « 

1 

2 

3 

4 

5 

Beaiing 

Degrees 

13 M 

d) <1) 

pq Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

9 

.988 

.156 

1.975 

.313 

2.963 

.469 

3.951 

.626 

4.938 

81 

9* 

.987 

.161 

1.974 

.321 

2.961 

.482 

3.948 

.643 

4.935 

80f 

9? 

.986 

.165 

1.973 

.330 

2.959 

.495 

3.945 

.660 

4.931 

80* 

9 f 

.986 

.169 

1.971 

.339 

2.957 

.508 

3.942 

.677 

4.928 

80* 

10 

.985 

.174 

1.970 

.347 

2.954 

.521 

3.939 

.695 

4.924 

80 

10 * 

.984 

.178 

1.968 

.356 

2.952 

.534 

3.936 

.712 

4.920 

79f 

10 * 

.983 

.182 

1.967 

.364 

2.950 

.547 

3.933 

.729 

4.916 

79* 

io* 

.982 

.187 

1.965 

.373 

2.947 

.560 

3.930 

.746 

4.912 

79* 

11 

.982 

.191 

1.963 

.3S2 

2.945 

.572 

3.927 

.763 

4.908 

79 

m 

.981 

.195 

1.962 

.390 

2.942 

.585 

3.923 

.780 

4.904 

78f 

Hi 

.980 

.199 

1.960 

.399 

2.940 

.598 

3.920 

.797 

4.900 

78* 

nf 

.979 

.204 

1.958 

.407 

2.937 

.611 

3.916 

.815 

4.895 

78* 

12 

.978 

.208 

1.956 

.416 

2.934 

.624 

3.913 

.832 

4.891 

78 

12 * 

.977 

.212 

1.954 

.424 

2.932 

.637 

3.909 

.849 

4.886 

77f 

12 * 

.976 

.216 

1.953 

.433 

2.929 

.649 

3.905 

.866 

4.881 

77* 

12 f 

.975 

.221 

1.951 

.441 

2.926 

.662 

3.901 

.883 

4.877 

77* 

13 

.974 

.225 

1.949 

.450 

2.923 

.675 

3.897 

.900 

4.872 

77 

131 

.973 

.229 

1.947 

.458 

2.920 

.688 

3.894 

.917 

4.867 

76f 

13* 

.972 

.233 

1.945 

.467 

2.917 

.700 

3.889 

.934 

4.862 

76* 

13f 

.971 

.238 

1.943 

.475 

2.914 

.713 

3.885 

.951 

4.857 

76* 

14 

.970 

.242 

1.941 

.484 

2.911 

.726 

3.881 

.968 

4.851 

76 

14i 

.969 

.246 

1.938 

.492 

2.908 

.738 

3.877 

.985 

4.846 

75! 

14i 

.968 

.250 

1.936 

.501 

2.904 

.751 

3.873 

1.002 

4.841 


14f 

.967 

.255 

1.934 

.509 

2.901 

.764 

3.868 

1.018 

4.835 

75* 

15 

.966 

.259 

1.932 

.518 

2.898 

.776 

3.864 

1.035 

4.830 

/ o 

151 

.965 

.263 

1.930 

.526 

2.894 

.789 

3.859 

1.052 

4.824 

74f 

151 

.964 

.267 

1.927 

.534 

2.891 

.802 

3.855 

1.069 

4.818 

74* 

15| 

.962 

.271 

1.925 

.543 

2.887 

.814 

3.850 

1 .0S6 

4.812 

74* 

16 

.961 

.276 

1.923 

.551 

2.884 

.827 

3.845 

1.103 

4.806 

74 

16* 

.960 

.280 

1.920 

.560 

2.880 

.839 

3.840 

1.119 

4.800 

73f 

16* 

.959 

.284 

1.918 

.568 

2.876 

.852 

3.835 

1.136 

4.794 

73* 

16f 

.958 

.288 

1.915 

.576 

2.873 

.865 

3.830 

1.153 

4.788 

73* 

17 

.956 

.292 

1.913 

.585 

2.869 

.877 

3.825 

1.169 

4.782 

73 

17* 

.955 

.297 

1.910 

.593 

2.865 

.890 

3.820 

1.186 

4.775 

72f 

17* 

.954 

.301 

1.907 

.601 

2.861 

.902 

3.815 

1.203 

4.769 

72* 

17f 

.952 

.305 

1.905 

.610 

2.857 

.915 

3.810 

1.220 

4.762 

72* 

18 

.951 

.309 

1.902 

.618 

2.853 

.927 

3.804 

1.236 

4.755 

72 

bo V* 
ST cd 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

be 

«rH 1 

ctJ be 

d; 










•S £> 










«U M 

ffi Q 


L 

2 

3 

4 

5 

PQ Q 








































USEFUL TABLES 17 


be V3 

c £ 

5 


6 


7 

8 


9 

c 84 












g s? 

CQ Q 










as bp 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

PQ Q 

9 

.782 

5.926 

.939 

6.914 

1.095 

7.902 

1.251 

8.889 

1.408 

81 

9? 

.804 

5.922 

.964 

6.909 

1.125 

7.896 

1.286 

8.883 

1.447 

80* 

9? 

.825 

5.918 

.990 

6.904 

1.155 

7.890 

1.320 

8.877 

1.485 

80* 

9* 

.847 

5.913 

1.016 

6.899 

1.185 

7.884 

1.355 

8.870 

1.524 

80* 

10 

.868 

5.909 

1.042 

6.894 

1.216 

7.878 

1.389 

8.863 

1.563 

80 

10 * 

.890 

5.904 

1.068 

6.888 

1.246 

7.872 

1.424 

8.856 

1.601 

79* 

10 * 

.911 

5.900 

1.093 

6.883 

1.276 

7.866 

1.458 

8.849 

1.640 

79* 

lOf 

.933 

5.895 

1.119 

6.877 

1.306 

7.860 

1.492 

8.842 

1.679 

79* 

11 

.954 

5.890 

1.145 

6.871 

1.336 

7.853 

1.526 

8.835 

1.717 

79 

m 

.975 

5.885 

1.171 

6.866 

1.366 

7.846 

1.561 

8.827 

1.756 

78* 

11 * 

.997 

5.880 

1.196 

6.859 

1.396 

7.839 

1.595 

8.819 

1.794 

78* 

Ilf 

1.018 

5.874 

1.222 

6.853 

1.425 

7.832 

1.629 

8.811 

1.833 

78* 

12 

1.040 

5.869 

1.247 

6.847 

1.455 

7.825 

1.663 

8.803 

1.871 

78 

12 * 

1.061 

5.863 

1.273 

6.841 

1.485 

7.818 

1.697 

8.795 

1.910 

77* 

12 * 

1.082 

5.858 

1.299 

6.834 

1.515 

7.810 

1.732 

8.787 

1.948 

77* 

12 f 

1.103 

5.852 1.324 

6.827 

1.545 

7.803 

1.766 

8.778 

1.986 

77* 

13 

1.125 

5.846 

1.350 

6.821 

1.575 

7.795 

1.800 

8.769 

2.025 

77 

13* 

1.146 

5.840 

1.375 

6.814 

1.604 

7.7S7 

1.834 

8.760 

2.063 

76* 

13* 

1.167 

5.834 

1.401 

6.807 

1.634 

7.779 

1.868 

8.751 

2.101 

76* 

13* 

1.188 

5.828 

1.426 

6.799 

1.664 

7.771 

1.902 

8.742 

2.139 

76* 

14 

1.210 

5.822 

1.452 

6.792 

1.693 

7.762 

1.935 

8.733 

2.177 

76 

14* 

1.231 

5.815 

1.477 

6.785 

1.723 

7.754 

1.969 

8.723 

2.215 

75* 

14* 

1.252 

5.809 

1.502 

6.777 

1.753 

7.745 

2.003 

8.713 

2.253 

75* 

14* 

1.273 

5.802 

1.528 

6.769 

1.782 

7.736 

2.037 

8.703 

2.291 

75* 

15 

1.294 

5.796 

1.553 

6.761 

1.812 

7.727 

2.071 

8.693 

2.329 

75 

15* 

1.315 

5.789 

1.578 

6.754 

1.841 

7.718 

2.104 

8.683 

2.367 

74* 

15* 

1.336 

5.782 

1.603 

6.745 

1.871 

7.709 

2.138 

8.673 

2.405 

74* 

15f 

1.357 

5.775 

1.629 

6.737 

1.900 

7.700 

2.172 

8.662 

2.443 

74* 

16 

1.378 

5.768 

1.654 

6.729 

1.929 

7.690 

2.205 

8.651 

2.481 

74 

16* 

1.399 

5.760 

1.679 

6.720 

1.959 

7.680 

2.239 

8.640 

2.518 

73* 

16* 

1.420 

5.753 

1.704 

6.712 

1.988 

7.671 

2.272 

8.629 

2.556 

73* 

16* 

1.441 

5.745 

1.729 

6.703 

2.017 

7.661 

2.306 

8.618 

2.594 

73* 

17 

1.462 

5.738 

1.754 

6.694 

2.047 

7.650 

2.339 

8.607 

2.631 

73 

17* 

1.483 

5.730 

1.779 

6.685 

2.076 

7.640 

2.372 

8.595 

2.669 

72* 

17* 

1.504 

5.722 

1.804 

6.676 

2.105 

7.630 

2.406 

8.583 

2.706 

72* 

17* 

1.524 

5.714 

1.829 

6.667 

2.134 

7.619 

2.439 

8.572 

2.744 

72* 

18 

1.545 

5.706 

1.854 

6.657 

2.163 

7.608 

2.472 

8.560 

2.781 

72 

be w 

c S 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

bo w 

Vh *-• 










•S3 

i-» <-• 

ce jo 

<D <L> 










d bp 

<D 

CQ Q 

5 

6 


7 


8 


9 


CQ Q 















































18 USEFUL TABLES 


c S 

l 

2 

3 

4 

5 

bo ft 

-c 2 










C *■« 

rt M 











« Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

« Q 

18 

.951 

.309 

1.902 

.618 

2.853 

.927 

3.804 

1.236 

4.755 

72 

18* 

.950 

.313 

1.899 

.626 

2.849 

.939 

3.799 

1.253 

4.748 

71* 

184 

.948 

.317 

1.897 

.635 

2.845 

.952 

3.793 

1.269 

4.742 

71* 

18* 

.947 

.321 

1.894 

.643 

2.841 

.964 

3.788 

1.286 

4.735 

71* 

19 

.946 

.326 

1.891 

.651 

2.837 

.977 

3.782 

1.302 

4.72S 

71 

194 

.944 

.330 

1.888 

.659 

2.832 

.989 

3.776 

1.319 

4.720 

70* 

194 

.943 

.334 

1.885 

.668 

2.828 

1.001 

3.771 

1.335 

4.713 

70* 

19f 

.941 

.338 

1.882 

.676 

2.824 

1.014 

3.765 

1.352 

4.706 

70* 

20 

.940 

.342 

1.879 

.684 

2.819 

1.026 

3.759 

1.368 

4.698 

70 

20 * 

.938 

.346 

1.876 

.692 

2.815 

1.038 

3.753 

1.384 

4.691 

69* 

20 | 

.937 

.350 

1.873 

.700 

2.810 

1.051 

3.747 

1.401 

4.683 

69* 

201 

.935 

.354 

1.870 

.709 

2.805 

1.063 

3.741 

1.417 

4.676 

69* 

21 

.934 

.358 

1.867 

.717 

2.801 

1.075 

3.734 

1.433 

4.668 

69 

21 * 

.932 

.362 

1.864 

.725 

2.796 

1.087 

3.728 

1.450 

4.660 

68 * 

21 * 

.930 

.367 

1.861 

.733 

2.791 

1.100 

3.722 

1.466 

4.652 

68 * 

21 f 

.929 

.371 

1.858 

.741 

2.786 

1.112 

3.715 

1.482 

4.644 

68 * 

22 

.927 

.375 

1.854 

.749 

2.782 

1.124 

3.709 

1.498 

4.636 

68 

22 J 

.926 

.379 

1.851 

.757 

2.777 

1.136 

3.702 

1.515 

4.628 

67* 

22 * 

.924 

.383 

1.848 

.765 

2.772 

1.148 

3.696 

1.531 

4.619 

67* 

23 1 

.922 

.387 

1.844 

.773 

2.767 

1.160 

3.689 

1.547 

4.611 

67* 

23 

.921 

.391 

1.841 

.781 

2.762 

1.172 

3.682 

1.563 

4.603 

67 

231 

.919 

.395 

1.838 

.789 

2.756 

1.184 

3.675 

1.579 

4.594 

66 * 

234 

.917 

.399 

1.834 

.797 

2.751 

1.196 

3.668 

1.595 

4.585 

66 * 

231 

.915 

.403 

1.831 

.805 

2.746 

1.208 

3.661 

1.611 

4.577 

66 * 

24 

.914 

.407 

1.827 

.813 

2.741 

1.220 

3.654 

1.627 

4.568 

66 

241 

.912 

.411 

1.824 

.821 

2.735 

1.232 

3.647 

1.643 

4.559 

65* 

244 

.910 

.415 

1.820 

.829 

2.730 

1.244 

3.640 

1.659 

4.550 

65* 

241 

.908 

.419 

1.816 

.837 

2.724 

1.256 

3.633 

1.675 

4.541 

65* 

25 

.906 

.423 

1.813 

.845 

2.719 

1.268 

3.625 

1.690 

4.532 

65 

251 

.904 

.427 

1.809 

.853 

2.713 

1.280 

3.618 

1.706 

4.522 

64* 

254 

.903 

.431 

1.805 

.861 

2.708 

1.292 

3.610 

1.722 

4.513 

64* 

251 

.901 

.434 

1.801 

.869 

2.702 

1.303 

3.603 

1.738 

4.503 

64* 

26 

.899 

.438 

1.798 

.877 

2.696 

1.315 

3.595 

1.753 

4.494 

64 

261 

.897 

.442 

1.794 

.885 

2.691 

1.327 

3.587 

1.769 

4.484 

63* 

264 

.895 

.446 

1.790 

.892 

2.685 

1.339 

3.580 

1.785 

4.475 

63* 

261 

.893 

.450 

1.786 

.900 

2.679 

1.350 

3.572 

1.800 

4.465 

63* 

27 

.891 

.454 

1.782 

.908 

2.673 

1.362 

3.564 

1.816 

4.455 

63 

M w 
a #} 

.3 <U 
>- b. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

» 8 
.5 

c3 bp 
<u <v 

m Q 

1 

2 

3 

4 

5 

& 60 
m Q 
















































USEFUL TABLES 


19 


he & 

c V, 

5 

6 

7 

8 

9 

1 

be & 

.3 <u 

Vh Jh 










.h a) 

n 

rt bo 










oj bo 

ra Q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

pq Q 

18 

1.545 

5.706 

1.854 

6.657 

2.163 

7.608 

2.472 

8.560 

2.781 

72 

18} 

1.566 

5.698 

1.879 

6.648 

2.192 

7.598 

2.505 

8.547 

2.818 

71} 

18} 

1.587 

5.690 

1.904 

6.638 

2.221 

7.587 

2.538 

8.535 

2.856 

715 

18} 

1.607 

5.682 

1.929 

6.629 

2.250 

7.575 

2.572 

8.522 

2.893 

71} 

19 

1.628 

5.673 

1.953 

6.619 

2.279 

7.564 

2.605 

8.510 

2.930 

71 

19} 

1.648 

5.665 

1.978 

6.609 

2.308 

7.553 

2.638 

8.497 

2.967 

70} 

194 

1.669 

5.656 

2.003 

6.598 

2.337 

7.541 

2.670 

8.484 

3.004 

70} 

19} 

1.690 

5.647 

2.028 

6.58S 

2.365 

7.529 

2.703 

8.471 

3.041 

70} 

20 

1.710 

5.638 

2.052 

6.578 

2.394 

7.518 

2.736 

8.457 

3.078 

70 

201 

1.731 

5.629 

2.077 

6.567 

2.423 

7.506 

2.769 

8.444 

3.115 

69} 

201 

1.751 

5.620 

2.101 

6.557 

2.451 

7.493 

2.802 

8.430 

3.152 

69} 

20 } 

1.771 

5.611 

2.126 

6.546 

2.480 

7.481 

2.834 

8.416 

3.189 

69} 

21 

1.792 

5.601 

2.150 

6.535 

2.509 

7.469 

2.867 

8.402 

3.225 

69 

21 } 

1.812 

5.592 

2.175 

6.524 

2.537 

7.456 

2.900 

8.388 

3.262 

68 } 

211 

1.833 

5.582 

2.199 

6.513 

2.566 

7.443 

2.932 

8.374 

3.299 

68 } 

21 f 

1.S53 

5.573 

2.223 

6.502 

2.594 

7.430 

2.964 

8.359 

3.335 

68 } 

22 

1.873 

5.563 

2.248 

6.490 

2.622 

7.417 

2.997 

8.345 

3.371 

68 

22 } 

1.893 

5.553 

2.272 

6.479 

2.651 

7.404 

3.029 

8.330 

3.408 

67} 

22 } 

1.913 

5.543 

2.296 

6.467 

2.679 

7.391 

3.061 

8.315 

3.444 

67} 

22 } 

1.934 

5.533 

2.320 

6.455 

2.707 

7.378 

3.094 

8.300 

3.480 

67} 

23 

1.954 

5.523 

2.344 

6.444 

2.735 

7.364 

3.126 

8.285 

3.517 

67 

23} 

1.974 

5.513 

2.368 

6.432 

2.763 

7.350 

3.158 

8.269 

3.553 

66 } 

23} 

1.994 

5.502 

2.392 

6.419 

2.791 

7.336 

3.190 

8.254 

3.589 

66 } 

23} 

2.014 

5.492 

2.416 

6.407 

2.819 

7.322 

3.222 

8.238 

3.625 

66 } 

24 

2.034 

5.481 

2.440 

6.395 

2.847 

7.308 

3.254 

S.222 

3.661 

66 

24} 

2.054 

5.471 

2.464 

6.382 

2.875 

7.294 

3.286 

8.206 

3.696 

65} 

24} 

2.073 

5.460 

2.488 

6.370 

2.903 

7.280 

3.318 

8.190 

3.732 

65} 

24f 

2.093 

5.449 

2.512 

6.357 

2.931 

7.265 

3.349 

8.173 

3.768 

65} 

25 

2.113 

5.438 

2.536 

6.344 

2.958 

7.250 

3.381 

8.157 

3.804 

65 

25} 

2.133 

5.427 

2.559 

6.331 

2.986 

7.236 

3.413 

8.140 

3.839 

64} 

25} 

2.153 

5.416 

2.583 

6.318 

3.014 

7.221 

3.444 

8.123 

3.875 

64} 

25} 

2.172 

5.404 

2.607 

6.305 

3.041 

7.206 

3.476 

8.106 

3.910 

64} 

26 

2.192 

5.393 

2.630 

6.292 

3.069 

7.190 

3.507 

8.089 

3.945 

64 

26} 

2.211 

5.381 

2.654 

6.278 

3.096 

7.175 

3.538 

8.072 

3.981 

63} 

26} 

2.231 

5.370 

2.677 

6.265 

3.123 

7.160 

3.570 

8.054 

4.016 

63} 

26} 

2.250 

5.358 

2.701 

6.251 

3.151 

7.144 

3.601 

8.037 

4.051 

63} 

27 

2.270 

5.346 

2.724 

6.237 

3.178 

7.128 

3.632 

8.019 

4.086 

63 

c % 
.5 <u 
n 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

.5 g 

vh H 

rt £0 










03 bo 

0) Oi 

CQ Q 

5 

6 


7 


8 

9 


(D O 

CQ Q 












































20 USEFUL TABLES 


C 3 

1 

2 


3 


4 


5 

bo w 

.5 ^ 

T ^ 










C ^ 

03 bo 
q; D 

d M 

<U <D 










ffl Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

m Q 

27 

.891 

.454 

1.782 

.90S 

2.673 

1.362 

3.564 

1.816 

4.455 

63 

27 j 

.889 

.458 

1.778 

.916 

2.667 

1.374 

3.556 

1.831 

4.445 

62! 

27 ! 

.887 

.462 

1.774 

.923 

2.661 

1.385 

3.548 

1.847 

4.435 

62! 

27| 

.885 

.466 

1.770 

.931 

2.655 

1.397 

3.540 

1.862 

4.425 

62! 

28 

.883 

.469 

1.766 

.939 

2.649 

1.408 

3.532 

1.878 

4.415 

62 

28! 

.881 

.473 

1.762 

.947 

2.643 

1.420 

3.524 

1.893 

4.404 

61! 

28 § 

.879 

.477 

1.758 

.954 

2.636 

1.431 

3.515 

1.909 

4.394 

61! 

28f 

.877 

.481 

1.753 

.962 

2.630 

1.443 

3.507 

1.924 

4.384 

61! 

29 

.875 

.485 

1.749 

.970 

2.624 

1.454 

3.498 

1.939 

4.373 

61 

29 j 

.872 

.4S9 

1.745 

.977 

2.617 

1.466 

3.490 

1.954 

4.362 

60! 

29! 

.870 

.492 

1.741 

.985 

2.611 

1.477 

3.481 

1.970 

4.352 

60! 

29f 

.868 

.496 

1.736 

.992 

2.605 

1.489 

3.473 

1.985 

4.341 

60! 

30 

.866 

.500 

1.732 

1.000 

2.598 

1.500 

3.464 

2.000 

4.330 

60 

30! 

.864 

.504 

1.728 

1.008 

2.592 

1.511 

3.455 

2.015 

4.319 

59! 

30 3 

.862 

.508 

1.723 

1.015 

2.585 

1.523 

3.447 

2.030 

4.308 

59! 

20! 

.859 

.511 

1.719 

1.023 

2.578 

1.534 

3.438 

2.045 

4.297 

59! 

31 

.857 

.515 

1.714 

1.030 

2.572 

1.545 

3.429 

2.060 

4.286 

59 

31! 

.855 

.519 

1.710 

1.038 

2.565 

1.556 

3.420 

2.075 

4.275 

58! 

31! 

.853 

.522 

1.705 

1.045 

2.558 

1.567 

3.411 

2.090 

4.263 

58! 

31! 

.850 

.526 

1.701 

1.052 

2.551 

1.579 

3.401 

2.105 

4.252 

58! 

32 

.848 

.530 

1.696 

1.060 

2.544 

1.590 

3.392 

2.120 

4.240 

58 

32! 

.846 

.534 

1.691 

1.067 

2.537 

1.601 

3.383 

2.134 

4.229 

57! 

32! 

.843 

.537 

1.687 

1.075 

2.530 

1.612 

3.374 

2.149 

4.217 

57! 

32! 

.841 

.541 

1.682 

1.082 

2.523 

1.623 

3.364 

2.164 

4.205 

57! 

33 

.839 

.545 

1.677 

1.089 

2.516 

1.634 

3.355 

2.179 

4.193 

57 

33! 

.836 

.548 

1.673 

1.097 

2.509 

1.645 

3.345 

2.193 

4.181 

56! 

33! 

.834 

.552 

1.668 

1.104 

2.502 

1.656 

3.336 

2.208 

4.169 

56! 

33! 

.831 

.556 

1.663 

1.111 

2.494 

1.667 

3.326 

2.222 

4.157 

56! 

34 

.829 

.559 

1.658 

1.118 

2.487 

1.678 

3.316 

2.237 

4.145 

56 

34! 

.827 

.563 

1.653 

1.126 

2.480 

1.688 

3.306 

2.251 

4.133 

55! 

34! 

.824 

.566 

1.648 

1.133 

2.472 

1.699 

3.297 

2.266 

4.121 

55! 

34! 

.822 

.570 

1.643 

1.140 

2.465 

1.710 

3.287 

2.280 

4.108 

55! 

35 

.819 

.574 

1.638 

1.147 

2.457 

1.721 

3.277 

2.294 

4.096 

55 

35! 

.817 

.577 

1.633 

1.154 

2.450 

1.731 

3.267 

2.309 

4.083 

54! 

35! 

.814 

.581 

1.628 

1.161 

2.442 

1.742 

3.257 

2.323 

4.071 

54! 

35! 

.812 

.584 

1.623 

1.168 

2.435 

1.753 

3.246 

2.337 

4.058 

54! 

36 

.809 

.588 

1.618 

1.176 

2.427 

1.763 

3.236 

2.351 

4.045 

54 

to w 

.5 jp 
H in. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

» 3 
•c S 

aj bfi 

<d <u 










ct to 

CQ Q 


1 


2 




4 

5 

m Q 








































USEFUL TABLES 21 


be & 

^ K 

5 

6 

7 

8 

9 

bo w 
c ^ 
•S CD 

a be 

<D Q> 










as be 

(1) <D 

CQ Q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

pq Q 

27 

2.270 

5.346 

2.724 

6.237 

3.178 

7.128 

3.632 

8.019 

4.086 

63 

27; 

2.289 

5.334 

2.747 

6.223 

3.205 

7.112 3.663 

8.001 

4.121 

62 J 

27* 

2.309 

5.322 

2.770 

6.209 

3.232 

7.09613.694 

7.983 

4.156 

62* 

27 i 

2.328 

5.310 

2.794 

6.195 

3.259 

7.080 

3.725 

7.965 

4.190 

62* 

28 

2.347 

5.298 

2.817 

6.181 

3.286 

7.064 

3.756 

7.947 

4.225 

62 

28 * 

2.367 

5.285 

2.840 

6.166 

3.313 

7.047 

3.787 

7.928 

4.260 

61; 

28* 

2.386 

5.273 

2.863 

6.152 

3.340 

7.031 

3.817 

7.909 

4.294 

61* 

28 f 

2.405 

5.260 

2.886 

6.137 

3.367 

7.014 

3.848 

7.891 

4.329 

61* 

29 

2.424 

5.248 

2.909 

6.122 

3.394 

6.997 

3.878 

7.872 

4.363 

61 

29; 

2.443 

5.235 

2.932 

6.107 

3.420 

6.980 

3.909 

7.852 

4.398 

60; 

29* 

2.462 

5.222 

2.955 

6.093 

3.447 

6.963 

3.939 

7.833 

4.432 

60* 

29 f 

2.481 

5.209 

2.977 

6.077 

3.474 

6.946 

3.970 

7.814 

4.466 

60* 

30 

2.500 

5.196 

3.000 

6.062 

3.500 

6.928 

4.000 

7.794 

4.500 

60 

30; 

2.519 

5.183 

3.023 

6.047 

3.526 

6.911 

4.030 

7.775 

4.534 

59; 

30* 

2.538 

5.170 

3.045 

6.031 

3.553 

6.893 

4.060 

7.755 

4.568 

59* 

30; 

2.556 

5.156 

3.068 

6.016 

3.579 

6.875 

4.090 

7.735 

4.602 

59* 

31 

2.575 

5.143 

3.090 

6.000 

3.605 

6.857 

4.120 

7.715 

4.635 

59 

31; 

2.594 

5.129 

3.113 

5.984 

3.631 

6.839 

4.150 

7.694 

4.669 

581 

31* 

2.612 

5.116 

3.135 

5.968 

3.657 

6.821 

4.180 

7.674 

4.702 

58* 

31 ; 

2.631 

5.102 

3.157 

5.952 

3.683 

6.803 

4.210 

7.653 

4.736 

58* 

32 

2.650 

5.088 

3.180 

5.936 

3.709 

6.784 

4.239 

7.632 

4.769 

58 

32; 

2.668 

5.074 

3.202 

5.920 

3.735 

6.766 

4.269 

7.612 

4.802 

57 ; 

32* 

2.686 

5.060 

3.224 

5.904 

3.761 

6.747 

4.298 

7.591 

4.836 

57* 

321 

2.705 

5.046 

3.246 

5.887 

3.787 

6.728 

4.328 

7.569 

4.869 

57* 

33 

2.723 

5.032 

3.268 

5.871 

3.812 

6.709 

4.357 

7.548 

4.902 

57 

331 

2.741 

5.018 

3.290 

5.854 

3.838 

6.690 

4.386 

7.527 

4.935 

56; 

33* 

2.760 

5.003 

3.312 

5.837 

3.864 

6.671 

4.416 

7.505 

4.967 

56* 

33 f 

2.778 

4.989 

3.333 

5.820 

3.889 

6.652 

4.445 

7.483 

5.000 

56* 

34 

2.796 

4.974 

3.355 

5.803 

3.914 

6.632 

4.474 

7.461 

5.033 

56 

34; 

2.814 

4.960 

3.377 

5.786 

3.940 

6.613 

4.502 

7.439 

5.065 

55; 

34* 

2.832 

4.945 

3.398 

5.769 

3.965 

6.593 

4.531 

7.417 

5.098 

55* 

341 

2.850 

4.930 

3.420 

5.752 

3.990 

6.573 

4.560 

7.395 

5.130 

55* 

35 

2.868 

4.915 

3.441 

5.734 

4.015 

6.553 

4.589 

7.372 

5.162 

55 

35; 

2.886 

4.900 

3.463 

5.716 

4.040 

6.533 

4.617 

7.350 

5.194 

54 f 

35* 

2.904 

4.885 

3.484 

5.699 

4.065 

6.513 

4.646 

7.327 

5.226 

54* 

35; 

2.921 

4.869 

3.505 

5.681 

4.090 

6.493 

4.674 

7.304 

5.258 

54* 

36 

2.939 

4.854 

3.527 

5.663 

4.115 

6.472 

4.702 

7.281 

5.290 

54 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

be cfl 
a £ 

.i: v 










0) 

a $o 

QJ <V 










PQ Q 

5 

6 

7 


8 

9 

PQ Q 














































22 USEFUL TABLES 


^ *h 

1 

2 

3 

4 

5 

be w 

.5 g 

oj be 
0) 0) 










oj bp 

PQ Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

CQ Q 

36 

.809 

.588 

1.618 

1.176 

2.427 

1.763 

3.236 

2.351 

4.045 

54 

36* 

.806 

.591 

1.613 

1.183 

2.419 

1.774 

3.226 

2.365 

4.032 

53 f 

363 

.804 

.595 

1.608 

1.190 

2.412 

1.784 

3.215 

2.379 

4.019 

53 § 

361 

.801 

.598 

1.603 

1.197 

2.404 

1.795 

3.205 

2.393 

4.006 

53 1 

37 

.799 

.602 

1.597 

1.204 

2.396 

1.805 

3.195 

2.407 

3.993 

53 

37 j 

.796 

.605 

1.592 

1.211 

2.388 

1.816 

3.184 

2.421 

3.9S0 

52* 

37 * 

.793 

.609 

1.587 

1.218 

2.380 

1.826 

3.173 

2.435 

3.967 

52 5 

37f 

.791 

.612 

1.581 

1.224 

2.372 

1.837 

3.163 

2.449 

3.953 

521 

38 

.788 

.616 

1.576 

1.231 

2.364 

1.847 

3.152 

2.463 

3.940 

52 

381 

.785 

.619 

1.571 

1.238 

2.356 

1.857 

3.141 

2.476 

3.927 

51 * 

381 

.783 

.623 

1.565 

1.245 

2.348 

1.868 

3.130 

2.490 

3.913 

51| 

381 

.780 

.626 

1.560 

1.252 

2.340 

1.878 

3.120 

2.504 

3.899 

51* 

39 

.777 

.629 

1.554 

1.259 

2.331 

1.888 

3.109 

2.517 

3.886 

51 

391 

.774 

.633 

1.549 

1.265 

2.323 

1.898 

3.098 

2.531 

3.S72 

50 * 

391 

.772 

.636 

1.543 

1.272 

2.315 

1.908 

3.086 

2.544 

3.858 

50* 

391 

.769 

.639 

1.538 

1.279 

2.307 

1.918 

3.075 

2.558 

3.844 

50 * 

40 

.766 

.643 

1.532 

1.286 

2.298 

1.928 

3.064 

2.571 

3.830 

50 

401 

.763 

.646 

1.526 

1.292 

2.290 

1.938 

3.053 

2.584 

3.816 

491 

401 

.760 

.649 

1.521 

1.299 

2.281 

1.948 

3.042 

2.598 

3.802 

49* 

401 

.758 

.653 

1.515 

1.306 

2.273 

1.958 

3.030 

2.611 

3.788 

49* 

41 

.755 

.656 

1.509 

1.312 

2.264 

1.968 

3.019 

2.624 

3.774 

49 

411 

.752 

.659 

1.504 

1.319 

2.256 

1.978 

3.007 

2.637 

3.759 

48* 

411 

.749 

.663 

1.498 

1.325 

2.247 

1.988 

2.996 

2.650 

3.745 

48* 

411 

.746 

.666 

1.492 

1.332 

2.238 

1.998 

2.984 

2.664 

3.730 

48* 

42 

.743 

.669 

1.486 

1.338 

2.229 

2.007 

2.973 

2.677 

3.716 

48 

421 

.740 

.672 

1.480 

1.345 

2.221 

2.017 

2.961 

2.689 

3.701 

47* 

421 

.737 

.676 

1.475 

1.351 

2.212 

2.027 

2,949 

2.702 

3.686 

47* 

421 

.734 

.679 

1.469 

1.358 

2.203 

2.036 

2.937 

2.715 

3.672 

47* 

43 

.731 

.682 

1.463 

1.364 

2.194 

2.046 

2.925 

2.728 

3.657 

47 

431 

.728 

.685 

1.457 

1.370 

2.185 

2.056 

2.913 

2.741 

3.642 

46* 

431 

.725 

.688 

1.451 

1.377 

2.176 

2.065 

2.901 

2.753 

3.627 

46* 

431 

.722 

.692 

1.445 

1.383 

2.167 

2.075 

2.889 

2.766 

3.612 

46* 

44 

.719 

.695 

1.439 

1.389 

2.158 

2.084 

2.877 

2.779 

3.597 

46 

441 

.716 

.698 

1.433 

1.396 

2.149 

2.093 

2.865 

2.791 

3.582 

45* 

441 

.713 

.701 

1.427 

1.402 

2.140 

2.103 

2.853 

2.804 

3.566 

45* 

441 

.710 

.704 

1.420 

1.408 

2.131 

2.112 

2.841 

2.816 

3.551 

45* 

45 

.707 

.707 

1.414 

1.414 

2.121 

2.121 

2.828 2.828 

3.536 

45 

be w 
c V. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 


C K 











rt bo 










PQ Q 

1 

2 

3 

4 

5 

CQ Q 









































USEFUL TABLES 


23 


bfi w 

c ^ 

.5 

U >-t 

5 

6 

7 

8 

9 

bo m 
c « 

rt bp 
<D QJ 



1 







3 3> 

m Q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

pq Q 

36 

2.939 

4.854 

3.527 

5.663 

4.115 

6.472 

4.702 

7.281 

5.290 

54 

361 

2.957 

4.839 

3.548 

5.645 

4.139 

6.452 

4.730 

7.258 

5.322 

53 f 

361 

2.974 

4.823 

3.569 

5.627 

4.164 

6.431 

4.759 

7.235 

5.353 

53 i 

36! 

2.992 

4.808 

3.590 

5.609 

4.188 

6.410 

4.787 

7.211 

5.385 

53 j 

37 

3.009 

4.792 

3.611 

5.590 

4.213 

6.389 

4.815 

7.188 

5.416 

53 

371 

3.026 

4.776 

3.632 

5.572 

4.237 

6.368 

4.842 

7.164 

5.448 

521 

37 1 

3.044 

4.760 

3.653 

5.554 

4.261 

6.347 

4.870 

7.140 

5.479 

52 5 

371 

3.061 

4.744 

3 673 

5.535 

4.286 

6.326 

4.898 

7.116 

5.510 

52 i 

38 

3.078 

4.728 

3.694 

5.516 

4.310 

6.304 

4.925 

7.092 

5.541 

52 

38 j 

3.095 

4.712 

3.715 

5.497 

4.334 

6.283 

4.953 

7.068 

5.572 

511 

381 

3.113 

4.696 

3.735 

5.478 

4.358 

6.261 

4.980 

7.043 

5.303 

511 

38 ! 

3.130 

4.679 

3.756 

5.459 

4.381 

6.239 

5.007 

7.019 

5.663 

511 

39 

3.147 

4.663 

3.776 

5.440 

4.405 

6.217 

5.035 

6.994 

5.664 

51 

39 1 

3.164 

4.646 

3.796 

5.421 

4.429 

6.195 

5.062 

6.970 

5.694 

50 i 

39 1 

3.180 

4.630 

3.816 

5.401 

4.453 

6.173 

5.089 

6.945 

5.725 

50 5 

39 f 

3.197 

4.613 

3.837 

5.382 

4.476 

6.151 

5.116 

6.920 

5.755 

50J 

40 

3.214 

4.596 

3.857 

5.362 

4.500 

6.128 

5.142 

6.894 

5.785 

50 

40j 

3.231 

4.579 

3.877 

5.343 

4.523 

6.106 

5.169 

6.869 

5.815 

49| 

40 2 

3.247 

4.562 

3.897 

5.323 

4.546 

6.083 

5.196 

6.844 

5.845 

49 j 

401 

3.264 

4.545 

3.917 

5.303 

4.569 

6.061 

5.222 

6.818 

5.875 

491 

41 

3.2S0 

4.528 

3.936 

5.283 

4.592 

6.038 

5.248 

6.792 

5.905 

49 

411 

3.297 

4.511 

3.956 

5.263 

4.615 

6.015 

5.275 

6.767 

5.934 

48| 

411 

3.313 

4.494 

3.976 

5.243 

4.638 

5.992 

5.301 

6.741 

5.964 

481 

41 f 

3.329 

4.476 

3.995 

5.222 

4.661 

5.968 

5.327 

6.715 

5.993 

481 

42 

3.346 

4.459 

4.015 

5.202 

4.684 

5.945 

5.353 

6.68S 

6.022 

48 

42j 

3.362 

4.441 

4.034 

5.182 

4.707 

5.922 

5.379 

6.662 

6.051 

47! 

421 

3.378 

4.424 

4.054 

5.161 

4.729 

5.898 

5.405 

6.635 

6.080 

471 

42 a 

3.394 

4.406 

4.073 

5.140 

4.752 

5.875 

5.430 

6.609 

6.109 

471 

43 

3.410 

4.388 

4.092 

5.119 

4.774 

5.851 

5.456 

6.582 

6.138 

47 

431 

3.426 

4.370 

4.111 

5.099 

4.796 

5.827 

5.481 

6.555 

6.167 

46! 

431 

3.442 

4.352 

4.130 

5.078 

4.818 

5.803 

5.507 

6.528 

6.195 

461 

43f 

3 458 

4.334 

4.149 

5.057 

4.841 

5.779 

5.532 

6.501 

6.224 

461 

44 

3.473 

4.316 

4.168 

5.035 

4.863 

5.755 

5.557 

6.474 

6.252 

46 

441 

3.489 

4.298 

4.187 

5.014 

4.885 

5.730 

5.582 

6.447 

6.280 

45! 

441 

3.505 

4.280 

4.206 

4.993 

4.906 

5.706 

5.607 

6.419 

6.308 

451 

44f 

3.520 

4.261 

4.224 

4.971 

4.928 

5.681 

5.632 

6.392 

6.336 

451 

45 

3.536 

4.243 

4.243 

4.950 

4.950 

5.657 

5.657 

6.364 

6.364 

45 

g> % 

.a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 a 

cs bp 










rt bp 

CQ Q 

5 

« 

6 


7 


8 

9 

cq Q 












































24 


MATHEMATICS 


MATHEMATICS 


SIMPLE PROPORTION, OR SINGLE RULE 

OF THREE 

A proportion is an expression of equality between equal 
ratios; thus, the ratio of 10 to 5 = the ratio of 4 to 2, and is 
expressed thus : 10 : 5 = 4:2. There are four terms in proportion; 
the first and last are the extremes and the second and third are 
the means. 

Quantities are in proportion by alternation when antecedent 
is compared with antecedent and consequent with consequent; 
thus, if 10:5 = 4:2, then 10:4 = 5:2. Quantities are in propor¬ 
tion by inversion when the antecedents are made consequents 
and the consequents antecedents; thus, if 10:5 = 4:2, then 
5:10 = 2:4. In any proportion, the product of the means will 
equal the product of the extremes, thus, if 10:5 = 4:2, then 
5X4 = 10X2. 

A mean proportional between two quantities equals the 
square root of their product; thus, a mean proportional between 
12 and 3 = the square root of 12X3, or 6. 

If the two means and one extreme of a proportion are given, 
the other extreme may be found by dividing the product of 
the means by the given extreme. Thus, 10:5 = 4:(), then 
(4X5)-PlO = 2, and the proportion is 10:5 = 4:2. If the two 
extremes and one mean are given, the other mean may be found 
by dividing the product of the extremes by the given mean. 
Thus, 10:() = 4:2, then (10X2)-f-4 = 5, and the proportion is 
10:5 = 4:2. 

Example. —If 6 men load 30 wagons of coal in a day, how 
many wagons will 10 men load? 

Solution. —They will evidently load more, so the second 
term of the proportion must be greater than the first. 

6:10 = 30:( ); then, (10X30)-^6 = 50. 


MA THEM AT ICS 


25 


PERCENTAGE 

Percentage means by or on the hundred. Thus, l% = jhs 
— .01, 3% = t^ = .03. 

To Find the Percentage, Having the Rate and the Base. 

Multiply the base by the rate expressed in hundredths; thus, 
6% of 1,930 is 1,930X.06 = 115.80. 

To Find the Amount, Having the Base and the Rate. —Mul¬ 
tiply the base by 1 plus the rate; thus, the amount of $1,930 
for 1 yr. at 6% is $1,930X 1.06 = $2,045.80. 

To Find the Base, Having the Rate and the Percentage. 
Divide the percentage by the rate; thus, if the rate is 6% and 
the percentage is 115.80, the base is 115.80-=.06= 1,930. 

To Find the Rate, Having the Percentage and the Base. 
Divide the percentage by the base; thus, if the percentage is 
115.80 and the base 1,930, the rate is 115.80-= 1,930 = .06, or 6%. 


FORMULAS 

The term formula , as used in mathematics and in technical 
books, may be defined as a rule in which symbols are used 
instead of words; in fact, a formula may be regarded as a 
shorthand method of expressing a rule. The signs used are 
the ordinary signs indicative of operations and the signs of 
aggregation; all of which are used in arithmetic. 

The use of formulas can best be shown by means of an 
example; therefore, the well-known rule for finding the horse¬ 
power of a steam engine will be taken. This rule may be stated 
as follows: 

Rule. —Divide the continued product of the mean effective 
pressure, in pounds per square inch, the length of the stroke, in 
feet, the area of the piston, in square inches, and the number of 
strokes per minute by 33,000: the result will be the horsepower. 

An examination of the rule will show that four quantities 
(viz., the mean effective pressure, the length of the stroke, 
the area of the piston, and the number of strokes) are multi¬ 
plied together, and the result is divided by 33,000. Hence, the 
rule might be expressed as follows: 



26 


MA THEM AT ICS 


_ mean effective pressure stroke 

orsepov er — ^ pounc j s per S q uare inch) (in feet) 

area of piston number of strokes 
(in square inches) A (per minute) 

This expression can be greatly shortened by representing 
each quantity by a single letter, thus representing horsepower 
by the letter H, the mean effective pressure, in pounds per 
square inch, by P, the length of the stroke in feet, by L, the 
area of the piston, in square inches, by A , the number of strokes 
per minute by N, and substituting these letters for the quan¬ 
tities that they represent, the following formula is obtained. 


PXLXAXN 

11 = - 

33,000 

The formula just given shows that a formula is really a 
shorthand method of expressing a rule. It is customary, 
however, to omit the sign of multiplication between two or 
more quantities when they are to be multiplied together, or 
between a number and a letter representing a quantity, it 
being always understood that when two letters are adjacent 
with no sign between them, the quantities represented by these 
letters are to be multiplied. Bearing this fact in mind, the 
formula just given can be further simplified to 


H = 


PLAN 

33,000 


The sign of multiplication, evidently, cannot be omitted 
between two or more numbers, as it would then be impossible 
to distinguish the numbers. 

Use of Formulas.—The area of any segment of a circle that 
is less than (or equal to) a semicircle is expressed by the 
formula irr 2 E c 


A = -■- (r — h), 

360 2 


in which A = area of segment; 

*- = 3.1416; 
r = radius; 

E = angle obtained by drawing lines from center to 
extremities of arc of segment; 
c = chord of segment; 
h = height of segment. 





MA THEM A TICS 


27 


Example. —What is the area of a segment whose chord is 
10 in. long, angle subtended by chord is 83.46°, radius, is 
7.5 in., and height of segment is 1.91 in.? 

Solution. —Applying the formula just given, 


A = 


3.1416X 7.5 2 X83.46 
360 


10 

—X (7.5-1.91) 
2 


= 40.968 — 27.95=13.018 sq. in., nearly 
The area of any triangle may be found by means of the 
following formula, 


in which 



2 


A = area; 


a, b, and c = lengths of sides. 

Example. —What is the area of a triangle whose sides are 
21 ft., 46 ft., and 50 ft. long? 

Solution. —In order to apply the formula, let a represent 
the side that is 21 ft. long; b, the side that is 50 ft. long; and c, 
the side that is 46 ft. long. Then, substituting, 


=fV 2i! -( 


212+502 —46 2 \ 2 


2X50 


> 


-f V 441 - ( 


441+2,500 -2 


100 


,116\2 / 

-1 = 25A/441 — 


/825\2 

\Too/ 


= 25 V441 -8.252 = 25. V441 - 68.0625 = 25 ^372.9375 


= 25X19.312 = 482.8 sq. ft., nearly 
These operations have been extended much further than was 
necessary; this was done in order to show the reader every 
step of the process. 

Rankine-Gordon Formula. —The Rankine-Gordon formula 
for determining the least load in pounds that will cause a long 
column to break is 


P = - 


SA 


1+9 


T- 

G'2 


in which P = load (pressure), in pounds; 

5 = ultimate strength of material composing column, 
in pounds per square inch; 














28 


MA THEM A TICS 


A = area of cross-section of column, in square inches; 
q = a factor (multiplier) whose value depends on 
shape of ends of column and on material com¬ 
posing column; 

1 = length of column, in inches; 

G = least radius of gyration of cross-section of column. 
Example. —What is the least load that will break a hollow 
steel column whose outside diameter is 14 in., inside diameter 
11 in., length 20 ft., and whose ends are flat? 

Solution. —For steel, 5=150,000, and q = - for flat- 

25,000 


ended steel columns; A = .7854(di 2 — cfe 2 ), di and do being the 
outside and inside diameters, respectively; ( = 20X12 = 240 in.; 


d\ 2 -f- di 2 

and G 2 =-—. Substituting these values in the formula, 


P = 


16 

150,000 X .7854(142 - 1l 2 ) 150,000 X 58.905 


-X- 


240 2 


1 + .1163 


25,000 14 2 +1 1 2 

16 


8,835.750 

1.1163 


= 7,915,211 lb. 


LOGARITHMS 

Logarithms are designed to diminish the labor of multiplica¬ 
tion and division, by substituting in their stead addition and 
subtraction. A logarithm is the exponent of the power to 
which a fixed number, called the base, must be raised to pro¬ 
duce a given number. The base of the common system is 10, 
and, as a logarithm is the exponent of the power to which the 
base must be raised in order to be equal to a given number, 
all numbers are to be regarded as powers of 10; hence, 

10°= 1, therefore logarithm of 1 = 0 

10 1= 10, therefore logarithm of 10= 1 
10 2 = 100, therefore logarithm of 100 = 2 
102= 1,000, therefore logarithm of 1,000 = 3 
10 4 = 10,000, therefore logarithm of 10,000 = 4 










MA THEM A TICS 


29 


The logarithms of numbers between 1 and 10 are less than 
unity, and are expressed as decimals. The logarithm of any 
number between 10 and 100 is more than 1 and less than 2, 
hence it is equal to 1 plus a decimal. Between 100 and 1,000 
it is equal to 2 plus a decimal, etc. 

The integral part of a logarithm is its characteristic, the deci¬ 
mal part is its mantissa. For example, the log of 67.7 is 
1.83059; the characteristic of this logarithm is 1 and the man¬ 
tissa is .83059. The characteristic of a logarithm is always 
1 less than the number of whole figures expressing that num¬ 
ber, and may be either negative or positive. The character¬ 
istic of the logarithm of 7 is 0; of 17 is 1; of 717 is 2; etc. The 
mantissa is always considered positive. 

To Find Logarithm of Any Number Between 1 and 100. 
Look on the first page of the table, along the column marked 
No., for the given number; opposite it will be found the loga¬ 
rithm with its characteristic. 

To Find Logarithm of Any Number of Three Figures. —Find 
the decimal in the first column to the right of the number; 
prefix to this the characteristic 2. Thus, the logarithm of 
327 is 2.51455. As the first two figures of the decimal are the 
same for several successive figures, they are only given where 
they change. Thus, the decimal part of the logarithm of 302 
is .48001. The first two figures remain the same up to 310, 
and are therefore to be supplied. 

To Find Logarithm of Any Number of Four Figures. —Look 
in the column headed No. for the first three figures, and then 
along the top of the page for the fourth figure. Down the 
column headed by the fourth figure, and opposite the first 
three, will be found the decimal part. To this prefix the char¬ 
acteristic 3. 

To Find Logarithm of Any Number of More Than Four 
Figures. —Place a decimal point after the fourth figure from 
the left, thus changing the number into an integer and a deci¬ 
mal. If the decimal part contains more than two figures, and 
its second figure is 5 or greater, add 1 to the first figure in the 
decimal. Find the mantissa of the first four figures, and sub¬ 
tract it from the next greater mantissa in the table. Under 
the heading P. P., find a column headed by the difference first 


30 


MA THEM A TICS 


found. Find in this column the number opposite the number 
corresponding to the first figure of the decimal, or the first 
figure increased by one, and add it to the mantissa already 
found for the first four figures of the given number. 

Example. —What is the logarithm of 234,567? 

Solution. —Placing a decimal point after the fourth figure 
from the left gives 2,345.67. The mantissa of 2,345 is .37014; 
the difference between .37014 and the next higher logarithm 
.37033 is 19. Add 1 to the first figure of the decimal 6, and 
in the column headed 19, under P. P., opposite 7, is found 13.3, 
which, added to the portion of the mantissa already found, 
.37014, gives .37027. The characteristic is 5, hence the loga¬ 
rithm is 5.37027. 

To Find Logarithm of Decimal Fraction. —Proceed according 
to the rules just given, except in regard to the characteristic. 
Where the number consists of a whole number and a decimal, 
the characteristic is 1 less than the whole number. Where it 
is a simple decimal, or when there are no ciphers between the 
decimal point and the first numerator, the characteristic is 
negative, and is expressed by 1, with a minus sign over it. 
Where there is one cipher between the decimal point and first 
numerator, the characteristic is 2, with a minus sign over it. 
Where there are 2 ciphers, the characteristic is 3, with a minus 
sign over it. Thus: 

The logarithm of 67.7 = 1.83059 

The logarithm of 6.77 =0.83059 

The logarithm of .677 =1.83059 

The logarithm of .0677 =2.83059 
The logarithm of .00677 = 3.83059 

The characteristic only is negative; the decimal part is 
positive. 

To Find Logarithm of Vulgar Fraction. —Subtract the loga¬ 
rithm of the denominator from the logarithm of the numerator; 
the difference is the logarithm of the fraction. 

Example. —Find logarithm of iV. 

Solution. — Log 4 = 0.60206 

Log 10= 1. 

1.60206 

1.60206 is the logarithm of .4. 


MA THEM A TICS 


31 


To Find Natural Number Corresponding to Any Logarithm. 

Look in the column headed 0 for the first two figures of the 
decimal part; the other four figures are to be looked for in the 
same or in one of the nine following columns. If they are 
exactly found, the number must be made to correspond with 
the characteristic by pointing off decimals or annexing ciphers. 

If the decimal portion cannot be found exactly, find the next 
lower logarithmT subtract it from the given logarithm, divide 
the difference by the difference between the next lower and the 
next higher logarithm, and annex the quotient to the natural 
number found opposite the lower logarithm. 

To Multiply by Logarithms. —Add the logarithms of the 
factors together; the sum will be the logarithm of their product. 

Example. —67.7 X .677 = ? 

Solution. — Log 67.7= 1.83059 
Log .677 = 1-83059 

1.66118 

1.66118 is the logarithm of 45.833 

To Divide by Logarithms. —Subtract the logarithm of the 
divisor from the logarithm of the dividend; the difference will 
be the logarithm of the quotient. 

Example. —Divide 67.7 by .0677. 

Solution. — Log 67.7 = 1.83059 
Log .0677 = 2-83059 
3.00000 

3 is the logarithm of 1,000 

To Square a Number by Logarithms. —Multiply the loga¬ 
rithm of the number by 2; the product will be the logarithm of 
the square of the number. 

Example. —Square .677. 

Solution. — Log .677 = 1-83059 

2 

1.66118 

1.66118 is the logarithm of .45833 

To Cube a Number by Logarithms. —Multiply the logarithm 
of the number by 3; the product will be the logarithm of the 
cube of the number. 





32 


MA THEM A TICS 


To Raise a Number to Any Power by Logarithms.—Multi¬ 
ply the logarithm of the number by 4, 5, 6, or 7, and the results 
will be the logarithms of the 4th, 5th, 6th, or 7th powers, 
respectively; thus, a number can readily be raised to any power 
required. 

To Extract Any Root of a Number by Logarithms.—Divide 

the logarithm of the number by the index of the root required; 
the quotient will be the logarithm of the required root. 

Example. —Find the square root of 625. 

Solution. — Logarithm of 625 = 2.79588 

2.795884-2 =1.39794 

1.39794 = logarithm of 25 

Therefore, the square root of 625 is 25. 

To Divide a Logarithm Having a Negative Characteristic. 

If the characteristic is evenly divisible by the divisor, divide 
in the usual manner and retain the negative sign of the char¬ 
acteristic in the quotient. If the negative characteristic is 
less than, or is not evenly divisible by, the divisor, add such 
a negative number to it as will make it evenly divisible, and 
prefix an equal positive number to the decimal part of the 
logarithm; then divide the increased negative characteristic 
by the divisor, to obtain the characteristic of the quotient 
desired. To obtain the decimal part of the quotient, divide 
the decimal part of the logarithm, with the positive number 
prefixed, in the usual manner. To this quotient prefix the 
negative characteristic already found, and this will be the 
quotient desired. Logarithms are particularly useful in those 
cases where the unknown quantity is an exponent, or when 
the exponent is a decimal. 

Example 1.—Divide 6.3246846 by 3. 

Solution.— 6.32468464-3 = 2.1582282 

Example 2.—Divide 14.3268472 by 9. 

Solution.— 14.32684724-9 = (14 + 4= 18) +(4+ .3268472) 

X 18+4.32684724-9= ^2.4807608 

Example 3. —Find ^677. 

Solution.— 

^677 = log .6774-5=1.8305784-5; 54-5 + 4.8305894-5 
= 1.9661178 = .9249 + 




MATHEMATICS 


33 


TABLE 

OF 

COMMON LOGARITHMS 

OF NUMBERS 

FROM 1 TO 10,000 


No. 

Log 

No. 

Log 

No. 

Log 

No. 

Log 

No. 

Log 

0 

— 

00 

20 

30 

103 

40 

60 

206 

60 

77 

815 

80 

90 

309 

1 

00 

000 

21 

32 

222 

41 

61 

278 

61 

78 

533 

81 

90 

849 

2 

30 

103 

22 

34 

242 

42 

62 

325 

62 

79 

239 

82 

91 

381 

3 

47 

712 

23 

36 

173 

43 

63 

347 

63 

79 

934 

83 

91 

908 

4 

60 

206 

24 

38 

021 

44 

64 

345 

64 

80 

618 

84 

92 

428 

5 

69 

897 

25 

39 

794 

45 

65 

321 

65 

81 

291 

85 

92 

942 

6 

77 

815 

26 

41 

497 

46 

66 

276 

66 

81 

954 

86 

93 

450 

7 

84 

510 

27 

43 

136 

47 

67 

210 

67 

82 

607 

87 

93 

952 

8 

90 

309 

2S 

44 

716 

48 

68 

124 

68 

83 

251 

88 

94 

448 

9 

95 

424 

29 

46 

240 

49 

69 

020 

69 

83 

885 

89 

94 

939 

10 

00 

000 

30 

47 

712 

50 

69 

897 

70 

84 

510 

90 

95 

424 

11 

04 

139 

31 

49 

136 

51 

70 

757 

71 

85 

126 

91 

95 

904 

12 

07 

918 

32 

50 

515 

52 

71 

600 

72 

85 

733 

92 

96 

379 

13 

11 

394 

33 

51 

851 

53 

72 

428 

73 

86 

332 

93 

96 

848 

14 

14 

613 

34 

53 

148 

54 

73 

239 

74 

86 

923 

94 

97 

313 

15 

17 

609 

35 

54 

407 

55 

74 

036 

75 

87 

506 

95 

97 

772 

16 

20 

412 

36 

55 

630 

56 

74 

819 

76 

88 

081 

96 

98 

227 

17 

23 

045 

37 

56 

820 

57 

75 

587 

77 

88 

649 

97 

98 

677 

18 

25 

527 

38 

57 

978 

58 

76 

343 

78 

89 

209 

98 

99 

123 

19 

27 

875 

39 

59 

106 

59 

77 

085 

79 

89 

763 

99 

99 

564 

20 

30 

103 

40 

60 

206 

60 

77 

815 

80 

90 

309 

100 

00 

000 
















































34 


MA THEM AT ICS 


COMMON LOGARITHMS. 


N. 

L. 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


P 

p. 


100 

00 000 

043 

087 

130 

173 

217 

260 

303 

346 

389 





101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 


44 

43 

42 

102 

860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 

1 

4.4 

4.3 

4.2 

103 

01 284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

2 

8.8 

8.6 

8.4 

104 

703 

745 

787 

828 

870 

912 

953 

995 

*036 

*078 

3 

13.2 

12.9 

12.6 

105 

02 119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17.6 

17.2 

16.8 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

5 

22.0 

21.5 

21.0 

107 

938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

6 

26.4 

25.8 

25.2 

108 

03 342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

7 

30.8 

30.1 

29.4 

109 

743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

8 

35.2 

34.4 

33.6 
















110 

04 139 

179 

218 

258 

297 

336 

376 

415 

454 

493 


oy.o 

00.7 

o 7 .o 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 


41 

40 

39 

112 

922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 

1 

4.1 

4.0 

3.9 

113 

05 308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

8.2 

8.0 

7.8 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

3 

12.3 

12.0 

11.7 

115 

06 070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

4 

16.4 

16.0 

15.6 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 

20.5 

20.0 

19.5 

117 

819 

856 

893 

930 

967 

*004 

*041 

*078 

*115 

*151 

6 

24.6 

24.0 

23.4 

118 

07 188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

7 

28.7 

28.0 

27.3 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

8 

32.8 

32.0 

31.2 

120 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 

9 

36.9 

36.0 

35.1 

121 

08 279 

314 

350 

386 

422 

458 

493 

529 

565 

600 


38 

37 

36 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

i 

3.8 

3.7 

3.6 

123 

991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

2 

7.6 

7.4 

7.2 

124 

09 342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

3 

11.4 

11.1 

10.8 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*003 

4 

15.2 

14.8 

14.4 

126 

10 037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

19.0 

18.5 

18.0 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

6 

22.8 

22.2 

21.6 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

7 

26.6 

25.9 

25.2 

129 

11 059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

8 

30.4 

29.6 

28.8 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

9 

34.2 

33.3 

32.4 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 


35 

34 

33 

132 

12 057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

i 

3.5 

3.4 

3.3 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

2 

7.0 

6.8 

6.6 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*001 

3 

10.5 

10.2 

9.9 

135 

13 033 

066 

098 

130 

162 

194 

226 

258 

290 

322 

4 

14.0 

13.6 

13.2 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

5 

17.5 

17.0 

16.5 

137 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

6 

21.0 

20.4 

19.8 

138 

988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

7 

24.5 

23.8 

23.1 

139 

14 301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

8 

28.0 

27.2 

26.4 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

~891 

9 

31.5 

30.6 

29.7 

141 

922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 


32 

31 

30 

142 

15 229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

1 

3.2 

3.1 

3.0 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

2 

6.4 

6.2 

6.0 

144 

836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

3 

9.6 

9.3 

9.0 

145 

16 137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

4 

12.8 

12.4 

12.0 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

5 

16.0 

15.5 

15.0 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

6 

19.2 

18.6 

18.0 

148 

17 026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

7 

22.4 

21.7 

21.0 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

8 

25.6 

24.8 

24.0 

150 

609 

638 

667 

696 

725 

754 

782 

811 

“840 

869 

9 

28.8 

27.9 

27.0 

N. 

L. 0 

1 

2 

3 

4 

5 

6 

/ 

8 

9 


p. 

P. 























































































































N. 

I5CT 

151 

152 

153 

154 

155 

156 

157 

158 

159 

160 

161 

162 

163 

164 

165 

166 

167 

168 

169 

170 

171 

172 

173 

174 

175 

176 

177 

178 

179 

180 

181 

182 

183 

184 

185 

186 

187 

188 

189 

190 

191 

192 

193 

194 

195 

196 

197 

198 

199 

200 

N. 


MATHEMATICS 


35 


Table—( Continued ). 


2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

667 

696 

725 

754 

782 

811 

840 

869 





955 

984 

*013 

*041 

*070 

*099 

*127 

*156 


29 

28 

241 

270 

298 

327 

355 

384 

412 

441 

l 

2.9 

2.8 

526 

554 

583 

611 

639 

667 

696 

724 

2 

5.8 

5.6 

808 

837 

865 

893 

921 

949 

977 

*005 

3 

8.7 

8.4 

089 

117 

145 

173 

201 

229 

257 

285 

4 

11.6 

11.2 

368 

396 

424 

451 

479 

507 

535 

562 

5 

14.5 

14.0 

645 

673 

700 

728 

756 

783 

811 

838 

6 

17.4 

16.8 

921 

948 

976 

*003 

*030 

*058 

*085 

*112 

7 

20.3 

19.6 

194 

222 

249 

276 

303 

330 

358 

385 

8 

23.2 

22.4 












466 

493 

520 

548 

575 

602 

629 

656 


^ O . 1 

4 i),A 

737 

763 

790 

817 

844 

871 

898 

925 


27 

26 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

1 

2.7 

2.6 

272 

299 

325 

352 

378 

405 

431 

458 

2 

5.4 

5.2 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.1 

7.8 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 

10.4 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13.5 

13.0 

324 

350 

376 

401 

427 

453 

479 

505 

6 

16.2 

15.6 

583 

608 

634 

660 

686 

712 

737 

763 

7 

18.9 

18.2 

840 

866 

891 

917 

943 

968 

994 

*019 

8 

21.6 

20.8 













096 

121 

147 

172 

198 

223 

249 

274 





350 

376 

401 

426 

452 

477 

502 

528 



Zb 

603 

629 

654 

679 

704 

729 

754 

779 

1 


2.5 

855 

880 

905 

930 

955 

980 

*005 

*030 

2 

5.0 

105 

130 

155 

180 

204 

229 

254 

279 

4 


7.5 

353 

378 

403 

428 

452 

477 

502 

527 

4 


10.0 

601 

625 

650 

674 

699 

724 

748 

773 

5 


12.5 

846 

871 

895 

920 

944 

969 

993 

*018 

6 


15.0 

091 

115 

139 

164 

188 

212 

237 

261 

7 


17.5 

334 

358 

382 

406 

431 

455 

479 

503 

8 

20.0 












575 

600 

624 

648 

672 

696 

720 

744 




816 

840 

864 

888 

912 

935 

959 

983 


Z4 

23 

055 

079 

102 

126 

150 

174 

198 

221 

1 

2.4 

2.3 

293 

316 

340 

364 

387 

411 

435 

458 

2 

4.8 

4.6 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7.2 

6.9 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9.6 

9.2 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 

12.0 

11.5 

231 

254 

277 

300 

323 

346 

370 

393 

6 

14.4 

13.8 

462 

485 

508 

531 

554 

577 

600 

623 

7 

16.8 

16.1 

692 

715 

738 

761 

784 

807 

830 

852 

8 

19.2 

18.4 









9 

a 

20.7 

921 

944 

967 

989 

*012 

*035 

*058 

*081 




149 

171 

194 

217 

240 

262 

285 

307 


22 

21 

375 

398 

421 

443 

466 

488 

511 

533 

1 

2.2 

2.1 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2 

825 

847 

870 

892 

914 

937 

959 

981 

3 

6.6 

6.3 

048 

070 

092 

115 

137 

159 

181 

203 

4 

8.8 

8.4 

270 

292 

314 

336 

358 

380 

403 

425 

5 

11.0 

10.5 

491 

513 

535 

557 

579 

601 

623 

645 

6 

13.2 

12.6 

710 

732 

754 

776 

798 

820 

842 

863 

7 

15.4 

14.7 

929 

951 

973 

994 

*016 

*038 

*060 

*081 

8 

17.6 

16.8 









y 

1 U W 

18.9 

146 

168 

190 

211 

233 

255 

276 

298 




2 

3 

4 

5 

6 

7 

8 

9 


P 

. P 








































































































N. 

!00 

201 

202 

203 

204 

205 

206 

207 

208 

209 

!!0 

211 

212 

213 

214 

215 

216 

217 

218 

219 

!2Q 

221 

222 

223 

224 

225 

226 

227 

228 

229 

130 

231 

232 

233 

234 

235 

236 

237 

238 

239 

!40 

241 

242 

243 

244 

245 

246 

247 

248 

249 

!50 

N. 


MA THEM A TICS 


Table—( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 



p 

P. 

146 

168 

190 

211 

233 

255 

276 

298 






363 

384 

406 

428 

449 

471 

492 

514 



22 

21 

578 

600 

621 

643 

664 

685 

707 

728 

l 


2.2 

2.1 

792 

814 

835 

856 

878 

899 

920 

942 

2 


4.4 

4.2 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

3 


6.6 

6.3 

218 

239 

260 

281 

302 

323 

345 

366 

4 


8.8 

8.4 

429 

450 

471 

492 

513 

534 

555 

576 

5 


11.0 

10.5 

639 

660 

681 

702 

723 

744 

765 

785 

6 


13.2 

12.6 

848 

869 

890 

911 

931 

952 

973 

994 

7 


15.4 

14.7 

056 

077 

098 

118 

139 

160 

181 

201 

8 


17.6 

16.8 

263 

284 

305 

325 

346 

366 

387 

408 

y 


19.8 

18.9 

469 

490 

510 

531 

552 

572 

593 

613 




20 

675 

695 

715 

736 

756 

777 

797 

818 


i 


2.0 

879 

899 

919 

940 

960 

980 

*001 

*021 


2 


4.0 

082 

102 

122 

143 

163 

183 

203 

224 


3 


6.0 

284 

304 

325 

345 

365 

385 

405 

425 


4 


8.0 

486 

506 

526 

546 

566 

586 

606 

626 


5 


10.0 

686 

706 

726 

746 

766 

786 

806 

826 


6 


12.0 

885 

905 

925 

945 

965 

985 

*005 

*025 


7 


14.0 

084 

104 

124 

143 

163 

183 

203 

223 


8 


16.0 

282 

301 

321 

341 

361 

380 

400 

420 


9 


18.0 

479 

498 

518 

537 

557 

577 

596 

616 




49 

674 

694 

713 

733 

753 

772 

792 

811 


i 


1.9 

869 

889 

908 

928 

947 

967 

986 

*005 


2 


3.8 

064 

083 

102 

122 

141 

160 

180 

199 


3 


5.7 

257 

276 

295 

315 

334 

353 

372 

392 


4 


7.6 

449 

468 

488 

507 

526 

545 

564 

583 


5 


9.5 

641 

660 

679 

698 

717 

736 

755 

774 


6 


11.4 

832 

851 

870 

889 

908 

927 

946 

965 


7 


13.3 

*021 

*040 

*059 

*078 

*097 

*116 

*135 

*154 


8 


15.2 

211 

229 

248 

267 

286 

305 

324 

342 


9 


17.1 

399 

418 

436 

455 

474 

493 

511 

530 




18 

586 

605 

624 

642 

661 

680 

698 

717 


i 


1.8 

773 

791 

810 

829 

847 

866 

884 

903 


2 


3.6 

959 

977 

996 

*014 

*033 

*051 

*070 

*088 


3 


5.4 

144 

162 

181 

199 

218 

236 

254 

273 


4 


7.2 

328 

346 

365 

383 

401 

420 

438 

457 


5 


9.0 

511 

530 

548 

566 

585 

603 

621 

639 


6 


10.8 

694 

712 

731 

749 

767 

785 

803 

822 


7 


12.6 

876 

894 

912 

931 

949 

967 

985 

*003 


8 


14.4 

057 

075 

093 

112 

130 

148 

166 

184 


9 


16.2 

238 

256 

274 

292 

310 

328 

346 

364 




17 

417 

435 

453 

471 

489 

507 

525 

543 


i 


1.7 

596 

614 

632 

650 

668 

686 

703 

721 


2 


3.4 

775 

792 

810 

828 

846 

863 

881 

899 


3 


5.1 

952 

970 

987 

*005 

*023 

*041 

*058 

*076 


4 


6.8 

129 

146 

164 

182 

199 

217 

235 

252 


5 


8.5 

305 

322 

340 

358 

375 

393 

410 

428 


6 


10.2 

480 

498 

515 

533 

550 

568 

585 

602 


7 


11.9 

655 

672 

690 

707 

724 

742 

759 

777 


8 


13.6 

829 

846 

863 

881 

898 

915 

933 

950 


9 


15.3 

2 

3 

4 

5 

6 

7 

8 

9 



P 

p 






























































MA THEM A TICS 


37 


Table —( Continued ). 


N . 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. p. 

250 

39 794 

811 

829 

846 

863 

881 

898 

915 

933 

950 



251 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 


18 

252 

40 140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

1 

1.8 

253 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

2 

3.6 

254 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

3 

5.4 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

4 

7.2 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

5 

9.0 

257 

993 

*010 

*027 

*044 

*061 

*078 

*095 

*111 

*128 

*145 

6 

10.8 

258 

41 162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

7 

12.6 

259 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

8 

14.4 













16.2 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 


261 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 


17 

262 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

1 

1.7 

263 

996 *012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

*144 

2 

3.4 

264 

42 160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 

5.1 

265 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

4 

6.8 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

5 

8.5 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

6 

10.2 

268 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

7 

11.9 

269 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

8 

9 

13.6 

15.3 

270 

43 136 152 

169 

185 

201 

217 

233 

249 

265 

281 

271 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 


16 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

1 

1.6 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

3.2 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

3 

4.8 

275 

933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

4 

6.4 

276 

44 091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

5 

8.0 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

6 

9.6 

278 

404 

420 

436 

451 

467 

483 

498 

514 529 

545 

7 

11.2 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

8 

Q 

12.8 

14.4 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 


281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 


15 

282 

45 025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

1 

1.5 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

2 

3.0 

284 

332 ! 347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4.5 

285 

484 500 

515 

530 

545 

561 

576 

591 

606 

621 

4 

6.0 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5 

7.5 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

6 

9.0 

288 

939 

954 

969 

984 

*000 

*015 

*030 

*045 *060 

*075 

7 

10.5 

289 

46 090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

8 

9 

12.0 

13.5 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 


291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 


14 

292 

538 

553 

568 

583 

598 

6)3 

627 

642 

657 

672 

1 

1.4 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

2 

2.8 

294 

835 

8501 

864 

879 

894 

909 

923 

938 

953 

967 

3 

4.2 

295 

982 

997 

*012 

*026 

*041 

*056 

*070 

* 085 , 

*100 

*114 

4 

5.6 

296 

47 129 

1441 

159 

173 

188 

202 

217 

232 ! 

246 ! 

261 

5 

7.0 

297 

276 

2901 

305 

319 

334 

349 

363 

378 

392 

407 

6 

8.4 

298 

422 

4361 

451 

465 

480 

494 

509 

524 

538 

553 

7 

9.8 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

8 

9 

11.2 

12.6 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p. 

p. 



















































































38 


MAT HEM A TICS 


Table—( Continued). 


N . 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

. P. 

300 

47 712 

727 

741 

756 

770 

784 

799 

813 

828 

842 



301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 



302 

48 001 

015 

029 

%4 

058 

073 

087 

101 

116 

130 


15 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 


304 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

1 

1.5 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

2 

3.0 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

3 

4.5 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

6.0 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

5 

7.5 

309 

996 

*010 

*024 

*038 

*052 

*066 

*080 

*094 

*108 

*122 

6 

7 

9.0 

10 5 

310 

49 136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

8 

12.0 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

9 

13.5 

312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 



313 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 



314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 



315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 


14 

316 

969 

982 

996 

*010 

*024 

*037 

*051 

*065 

*079 

*092 

1 

1.4 

317 

50 106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

2 

2.8 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

3 

4.2 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

4 

5.6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

5 

6 

7.0 

8 4 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9.8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

11.2 

323 

920 

934 

947 

961 

974 

987 

*001 

*014 

*028 

*041 

9 

12.6 

324 

51 055 

068 

081 

095 

108 

121 

135 

148 

162 

175 



325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 



326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 



327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 


13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

1 

1 3 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 

2.6 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

3 

3.9 

331 

983 

996 

*009 

*022 

*035 

*048 

*061 

*075 

*088 

*101 


0.2 

332 

52 114 

127 

140 

153 

166 

179 

192 

205 

218 

231 


0.0 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 



334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 



335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 



336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 



337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 



338 

892 

905 

917 

930 

943 

956 

969 

982 

994 

*007 



339 

53 020 

033 

046 

058 

071 

084 

097 

110 

122 

135 


12 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

1 

1.2 

341 

275 

288 

301 

314 

326 

339 

352 , 

364 

377 

390 

2 

2.4 

342 

403 

415 

428 

441 

453 

466 

479 , 

491 

504 

517 

3 

3.6 

343 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

4 

4.8 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 | 

769 

5 

6.0 

345 

782 

794 

807 

820 

832 

845 

857 

870 

8821 

895 

6 

7.2 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*008 

*020 

7 

8.4 

347 

54 033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

8 

9.6 

348 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

9 

10.8 

349 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 



350 

407 

419 

432 

444 

456 

469 

481 

494 

“506 

518 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P. 









































































































MATHEMATICS 


39 


Table— ( Continued). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

¥ 

. p. 

350 

54 407 

419 

432 

444 

456 

469 

481 

494 

506 

518 



351 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 



352 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 



353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 


13 

354 

900 

913 

925 

937 

949 

962 

974 

986 

998 

*011 

1 

1.3 

355 

55 023 

035 

047 

060 

072 

' 084 

096 

108 

121 

133 

2 

2.6 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

3 

3.9 

357 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

4 

5.2 

358 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

5 

8.5 

359 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

6 

7.8 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

8 

10.4 

361 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 

9 

11.7 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 



363 

991 

*003 

*015 

*027 

*038 

*050 

*062 

*074 

*086 

“098 



364 

56 110 

122 

134 

146 

158 

170 

182 

194 

205 

217 



365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 


12 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

1 

1.2 

367 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

2 

2.4 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

3.6 

369 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

4 

4.8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

5 

6 

6.0 

7.2 

371 

937 

949 

961 

972 

984 

996 

*008 

*019 

*031 

*043 

7 

8.4 

372 

57 054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

8 

9.6 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

9 

10.8 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 



375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 



376 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 



377 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 


II 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

1 

1.1 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 

2.2 

380 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

3 

3.3 

381 

58 092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

5 

5 5 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

6 

6 6 

383 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

7 

7 7 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

8 

8 8 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

9 

9 9 

386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 



387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 



388 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 



389 

995 

*006 

*017 

*028 

*040 

*051 

*062 

*073 

*084 

*095 


10 

390 

59 106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

i 

1.0 

391 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

2 

2.0 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 


o.O 

393 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

4 

4.0 

394 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 


o.O 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

6 

6.0 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

7 

7.0 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

8 

8.0 

398 

988 

999 

*010 

*021 

*032 

*043 

*054 

*065 

*076 

*086 

9 

9.0 

399 

60 097 

108 

119 

130 

141 

152 

163 

173 

184 

195 



400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 



N. 

L.O i 

1 ] 

2 

3 

4 

5 

6 

7 

8 

9 

p 

p. 

















































































































40 


MA THEM A TICS 


Table —( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

F 

. p. 

400 

60 206 

217 

228 

239 

249 

260 

271 

282 

293 

304 



401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 



402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 



403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 



404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 



405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 



406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 


ii 

407 

959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 

l 

i.i 

408 

61 066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

2 

2.2 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

3 

3.3 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

4 

t \ 

4.4 
k ^ 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

6 

6 (> 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

7 

7 7 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

g 

ft ft 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

9 

99 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 



416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 



417 

62 014 

024 

034 

045 

055 

066 

076 

086 

097 

107 



418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 



419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 



420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 



421 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 


10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

i 

1.0 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

2 

2.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

3.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

4 

4.0 

426 

941 

951 

961 

972 

982 

992 

*002 

*012 

*022 

*033 

5 

5.0 

427 

63 043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

6 

6.0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

7 

7.0 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

8 

8.0 














430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 



431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 



432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 



433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 



434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 



435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 



436 

949 

959 

969 

979 

988 

998 

*008 

*018 

*028 

*038 


q 

437 

64 048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

1 

0 9 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

2 

1 8 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

3 

2.7 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 


4 

3.6 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

5 

4.5 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

6 

5.4 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

7 

6.3 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 


7.2 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 


6.1 

446 

933 

943 

953 

963 

972 

982 

992 

*002 

*011 

*021 



447 

65 031 

040 

050 

060 

070 

079 

089 

099 

108 

118 



448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 



449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 



450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

~408 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

. p. 



















































































































50 

451 

452 

453 

454 

455 

456 

457 

458 

459 

60 

461 

462 

463 

464 

465 

466 

467 

468 

469 

70 

471 

472 

473 

474 

475 

476 

477 

478 

479 

80 

481 

482 

483 

484 

485 

486 

487 

488 

489 

90 

491 

492 

493 

494 

495 

496 

497 

498 

499 

00 

N . 


MA THEM A TICS 


41 


Table— ( Continued ). 


2 

3 

4 

5 

6 

7 

8 

9 

341 

350 

360 

369 

379 

389 

398 

408 

437 

447 

456 

466 

475 

485 

495 

504 

533 

543 

552 

562 

571 

581 

591 

600 

629 

639 

648 

658 

667 

677 

686 

696 

725 

734 

744 

753 

763 

772 

782 

792 

820 

830 

839 

849 

858 

868 

877 

887 

916 

925 

935 

944 

954 

963 

973 

982 

*011 

*020 

*030 

*039 

*049 

*058 

*068 

*077 

106 

115 

124 

134 

143 

153 

162 

172 

200 

210 

219 

229 

238 

247 

257 

266 

295 

304 

314 

323 

332 

342 

351 

361 

389 

398 

408 

417 

427 

436 

445 

455 

483 

492 

502 

511 

521 

530 

539 

549 

577 

586 

596 

605 

614 

624 

633 

642 

671 

680 

689 

699 

708 

717 

727 

736 

764 

773 

783 

792 

801 

811 

820 

829 

857 

867 

876 

885 

894 

904 

913 

922 

950 

960 

969 

978 

987 

997 

*006 

*015 

043 

052 

062 

071 

080 

089 

099 

108 

136 

145 

154 

164 

173 

182 

191 

201 

228 

237 

247 

256 

265 

274 

284 

293 

321 

330 

339 

348 

357 

367 

376 

385 

413 

422 

431 

440 

449 

459 

468 

477 

504 

514 

523 

532 

541 

550 

560 

569 

596 

605 

614 

62 4 

633 

642 

651 

660 

688 

697 

706 

715 

724 

733 

742 

752 

779 

788 

797 

806 

815 

825 

834 

843 

870 

879 

888 

897 

906 

916 

925 

934 

961 

970 

979 

988 

997 

*006 

*015 

*024 

052 

061 

070 

079 

088 

097 

106 

115 

142 

151 

160 

169 

178 

187 

196 

205 

233 

242 

251 

260 

269 

278 

287 

296 

323 

332 

341 

350 

359 

368 

377 

386 

413 

422 

431 

4 40 

449 

458 

467 

476 

502 

511 

520 

529 

538 

547 

556 

565 

592 

601 

610 

619 

628 

637 

646 

655 

681 

690 

699 

708 

717 

726 

735 

744 

771 

780 

789 

797 

806 

815 

824 

833 

860 

869 

878 

886 

895 

904 

>913 

922 

949 

958 

966 

975 

984 

993 

*002 

*011 

037 

046 

055 

064 

073 

082 

090 

099 

126 

135 

144 

152 

161 

170 

179 

188 

214 

223 

232 

241 

249 

258 

267 

276 

302 

311 

320 

329 

338 

346 

355 

364 

390 

399 

408 

417 

425 

434 

443 

452 

478 

487 

496 

504 

513 

522 

531 

539 

566 

574 

583 

592 

601 

609 

618 

627 

653 

662 

671 

679 

688 

697 

705 

714 

740 

749 

758 

767 

775 

784 

793 

801 

827 

836 

845 

854 

862 

871 

880 

888 

914 

923 

932 

940 

949 

958 

966 

975 

2 

3 

4 

5 

6 

7 

8 

9 


P. P. 


1 

2 

3 

4 

5 

6 

7 

8 
9 


10 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

9.0 


1 

2 

3 

4 

5 

6 

7 

8 
9 


9 

0.9 

1.8 

2.7 

3.6 

4.5 

5.4 

6.3 

7.2 

8.1 


1 

2 

3 

4 

5 

6 

7 

8 
9 


8 

0.8 

1.6 

2.4 

3.2 
4.0 
4.8 
5.6 

6.4 

7.2 


P. P. 



















































































42 


MATHEMATICS 


Table —( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

. p. 

500 

69 897 

906 

914 

923 

932 

940 

949 

958 

966 

975 



501 

984 

992 

*001 

*010 

*018 

*027 

*036 

*044 

*053 

*062 



502 

70 070 

079 

088 

096 

105 

114 

122 

131 

140 

148 



503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 



504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 



505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 



506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 


9 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

1 

0 9 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

2 

1 8 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

3 

2.7 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

4 

3.6 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

5 

4.5 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

*003 



513 

71 012 

020 

029 

037 

046 

054 

063 

071 

079 

088 



514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 



515 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 



516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 



517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 



518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 



519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 



520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 



521 

684 

692 

700 

709 

717 

725 

734 

742 

750 

“759 


8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

1 

0.8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2 

1.6 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

3 

2.4 

525 

72 016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

4 

3.2 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4.0 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

6 

4.8 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

7 

5.6 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

8 

6.1 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

9 

7.2 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 



532 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 



533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 



534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 



535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 



536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 


7 

537 

997 

*006 

*014 

*022 

*030 

*038 

*046 

*054 

*062 

*070 



538 

73 078 

086 

094 

102 

in 

119 

127 

135 

143 

151 



539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

3 

2.1 

540 

239 

247 

255 

263 

272 

280 

288 

296 

“304 

^£2 

4 

2.8 

541 

320 

328 

336 

344 

352 

360 

368 

“376 

“384 

“392 

5 

3.5 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

6 

4.2 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

7 

4.9 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

8 

5.6 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 

9 

6.3 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 



547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 



548 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 



549 

957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 



550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

* 

9 

p 

p. 




















































































MA THEM A TICS 


43 


Table— ( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. p. 

550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 



552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 



553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 



554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 



555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 



556 

507 

515 

523 

631 

539 

547 

554 

562 

570 

578 



557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 



568 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 



559 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 



560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 


8 

561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 


562 

974 

981 

989 

997 

*005 

*012 

*020 

*028 

*035 

*043 

1 

0.8 

563 

75 051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

2 

1.6 

564 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

3 

2.4 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

4 

3.2 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

5 

4.0 

567 

358 

366 

374 

381 

289 

397 

404 

412 

420 

427 

6 

4.8 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

7 

o.o 

569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

o 

9 

6.4 

7.2 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 


571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 



572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 



573 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 



574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 



575 

967 

974 

982 

989 

997 

*005 

*012 

*020 

*027 

*035 



576 

76 042 

050 

057 

065 

072 

080 

087 

095 

103 

110 



577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 



578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 



579 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 



580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 



581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 


7 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

i 

0.7 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

2 

1.4 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

3 

2.1 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

4 

2.8 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

5 

3.5 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

6 

4.2 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

*004 

7 

4.9 

589 

77 012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

8 

5.6 












9 

6.3 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 


591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 



592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 



593 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 



594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 



595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 



596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 



597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 



598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 



599 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 



600 

815 

822 

830 

837 

844 

851 

«59 

866 l 

873 

880 



N. 

L.O 

1 

2 

3 

4 

5 

6 1 

7 

8 | 

9 

p. p. 
























































































































44 


MATHEMATICS 


Table —( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. P. 

600 

77 815 

822 

830 

837 

844 

851 

859 

866 

873 

880 



601 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 



602 

960 

967 

974 

981 

988 

996 

*003 

*010 

*017 

*025 



603 

78 032 

039 

046 

053 

061 

068 

075 

082 

089 

097 



604 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 



605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 



606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 


8 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

1 

0.8 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

2 

1.6 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

3 

2.4 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

4 

3.2 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 



612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 



613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 



614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

Q 


615 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 



616 

958 

965 

972 

979 

986 

993 

*000 

*007 

*014 

*021 



617 

79 029 

036 

043 

050 

057 

064 

071 

078 

085 

092 



618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 



619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 



620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 



621 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 


7 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

l 

0.7 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

2 

1.4 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

3 

2.1 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

4 

2.8 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

5 

3.5 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

6 

4.2 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

7 

4.9 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

8 

5.6 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

9 

6.8 

631 

80 003 

010 

017 

024 

030 

037 

044 

051 

058 

065 



632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 



633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 



634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 



635 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 



636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 



637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

\ 


638 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 



639 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

3 

1.8 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

4 

2.4 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

5 

3.0 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

6 

3.6 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

7 

4.2 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

8 

4.8 

645 

956 

963 

969 

976 

983 

990 

996 

*003 

*010 

*017 

9 

5.4 

646 

81 023 

030 

037 

043 

050 

057 

064 

070 

077 

084 



647 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 



648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 



649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 



650 

291 

298 

305 

311 

318 

325 

331 

338 

”345 

351 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

. p. 














































































































MA THEM A TICS 


45 


Table— ( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

F 

\ P. 

650 

81 291 

298 

305 

311 

318 

325 

331 

338 

345 

351 



651 

358 

365 

371 

378 

385 

391 

398 

405 

411 

hns 



652 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 



653 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 



654 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 



655 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 



656 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 



657 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 



658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 



659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 



660 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 



661 

82 020 

027 

033 

040 

046 

053 

060 

066 

073 

079 


7 

662 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

1 

0.7 

663 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

2 

1.4 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

3 

2 .1 

665 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

4 

2.8 

666 

347 

354 

360 

367 

373 

S 80 

387 

393 

400 

406 

5 

3.5 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

6 

4.2 

668 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

7 

4.9 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

8 

9 

5.6 

6.3 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 


671 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 



672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 



673 

802 

808 

814 

821 

827 

834 

840 

847 853 

860 



674 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 



675 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 



676 

995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 



677 

83 059 

065 

072 

078 

085 

091 

097 

104 

110 

117 



678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 



679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 



680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 



681 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 


6 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

i 

0.6 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

2 

1.2 

684 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

3 

1.8 

685 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

4 

2.4 

686 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

5 

3.0 

687 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

6 

3.6 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

7 

4.2 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

8 

4.8 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 



691 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 



692 

84 011 

017 

023 

029 

036 

042 

048 

055 

061 

067 



693 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 



694 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 



695 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 



696 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 



697 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 



698 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 



699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 



700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. P. 























































































46 


MA THEM A TICS 


Table— ( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

ir 

\ p. 

700 

84 510 

516 

522 

528 

535 

541 

547 

553 

559 

566 



701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 



702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 



703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 



704 

757 

763 

770 

776 

782 

788 

794 

| 800 

807 

813 



705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 



706 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 


7 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

1 

0 7 

708 

85 003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

2 

1.4 

709 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

3 

2.1 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

4 

2.8 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

0 

o.D 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 



713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 



714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 



715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 



716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 



717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 



718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 



719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 



720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 



721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 


6 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

1 

0.6 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

2 

1.2 

724 

974 

980 

986 

992 

998 

*004 

*010 

*016 

•'022 

*028 

3 

1.8 

725 

86 034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

4 

2.4 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

5 

3.0 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

6 

3.6 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

7 

4.2 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

8 

4.8 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

9 

5.4 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 



732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 



733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 



734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 



735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 



736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 


6 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

1 

0 5 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

2 

1 0 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

3 

1.5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

“970 

~976 

4 

2.0 

741 

982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

5 

2.5 

742 

87 040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

6 

3.0 

743 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

7 

3.5 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

8 

4.0 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

9 

4.5 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 



747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 



748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 



749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 



750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

“558 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p. p. 








































































































MA THEM A TICS 


47 


Table —( Continued ). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. P. 

750 

87 506 

512 

518 

523 

529 

535 

541 

547 

552 

558 



751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

”616 



752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 



753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 



754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 



755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 



756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 



757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 



758 

967 

973 

978 

984 

990 

996 

*001 

*007 

*013 

*018 



759 

88 024 

030 

036 

041 

047 

053 

058 

064 

070 

076 



760 

081 

00 1 
PJ 

093 

098 

104 

110 

116 

121 

127 

133 



761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 


6 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

l 

0.6 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

2 

1.2 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

3 

1.8 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

4 

2.4 

766 

423 

429 

434 

440 

446 

451 

457 

463 

1 468 

474 

5 

3.0 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

6 

3.6 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

7 

4.2 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

8 

4.8 

5.4 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 


771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 



772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 



773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 



774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 



775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 



776 

986 

992 

997 

*003 

*009 

*014 

*020 

*025 

*031 

*037 



777 

89 042 

048 

053 

059 

064 

070 

076 

081 

087 

092 



778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 



779 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 



780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 



781 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 


5 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

i 

0.5 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

2 

1.0 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

3 

1.5 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

4 

2.0 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

5 

2.5 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

6 

3.0 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

7 

3.5 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

8 

4.0 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

« 9 

4.5 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 



792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 



793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 



794 

982 

988 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 



795 

90 037 

042 

048 

053 

059 

064 

069 

075 

080 

086 



796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 



797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 



798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 



799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 



800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 



N. 

L.O 

1 

2 

3 

4 

5 1 

6 1 

7 

8 1 

9 

p 

p . 





















































































N. 

soo 

801 

802 

803 

804 

805 

806 

807 

808 

809 

110 

811 

812 

813 

814 

815 

816 

817 

818 

819 

1 2 0 

821 

822 

823 

824 

825 

826 

827 

828 

829 

30 

831 

832 

833 

834 

835 

836 

837 

838 

639 

40 

841 

842 

843 

844 

845 

846 

847 

848 

849 

50 

N. 


MA THEM A TICS 


Table —( Continued). 


L.O 

1 1 

2 

3 

4 

5 

6 

7 

8 

9 

I 

\ P. 

90 309 

314 

320 

325 

331 

336 

342 

347 

352 

358 



363 

369 

374 

380 

385 

390 

396 

401 

407 

412 



417 

423 

428 

434 

439 

445 

450 

455 

461 

466 



472 

477 

482 

488 

493 

499 

504 

509 

515 

520 



526 

531 

536 

542 

547 

553 

558 

563 

569 

574 



580 

585 

590 

596 

601 

607 

612 

617 

623 

628 



634 

639 

644 

650 

655 

660 

666 

671 

677 

682 



687 

693 

698 

703 

709 

714 

720 

725 

730 

736 



741 

747 

752 

757 

763 

768 

773 

779 

784 

789 



795 

800 

806 

811 

816 

822 

827 

832 

838 

843 



849 

854 

859 

865 

870 

875 

881 

886 

891 

897 


6 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 

1 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*004 

0.6 

91 009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

1.2 

062 

068 

073 

078 

084 

089 

094 

100 

105 

110 

3 

1.8 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

4 

2.4 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

5 

3.0 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6 

3.6 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

7 

4.2 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 

8 

9 

4.8 

K 4 . 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 



434 

440 

445 

450 

455 

461 

466 

471 

477 

482 



487 

492 

498 

503 

508 

514 

519 

524 

529 

535 



540 

545 

551 

556 

561 

566 

572 

577 

582 

587 



593 

598 

603 

609 

614 

619 

624 

630 

635 

640 



645 

651 

656 

661 

666 

672 

677 

682 

687 

693 



698 

703 

709 

714 

719 

724 

730 

735 

740 

745 



751 

756 

761 

766 

772 

777 

782 

787 

793 

798 



803 

808 

814 

819 

824 

829 

834 

840 

845 

850 



855 

861 

866 

871 

876 

882 

887 

892 

897 

903 



908 

913 

918 

924 

929 

934 

939 

944 

950 

955 



960 

965 

971 

976 

981 

986 

991 

997 

*002 

* 7)07 


5 

92 012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

i 

0.5 

065 

070 

075 

080 

085 

091 

096 

101 

106 

111 

2 

1.0 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

3 

1.5 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

4 

2.0 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

2.5 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

6 

3.0 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

7 

3.5 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

8 

4.0 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

9 

4.5 

480 

485 

490 

495 

500 

505 

511 

516 

“521 

526 



531 

536 

542 

547 

552 

557 

562 

567 

572 

578 



583 

588 

593 

598 

603 

609 

614 

619 

624 

629 



634 

639 

645 

650 

655 

660 

665 

670 

675 

681 



686 

691 

696 

701 

706 

711 

716 

722 

727 

732 



737 

742 

747 

752 

758 

763 

768 

773 

778 

783 



788 

793 

799 

804 

809 

814 

819 

824 

829 

334 



840 

845 

850 

855 

860 

865 

870 

875 

881 

886 



891 

896 

901 

906 

911 

916 

921 

927 

932 

937 



942 

947 

952 

957 

962 

967 

973 

978 

983 

”988 



L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. p. 



























































MATHEMATICS 


49 


Table —( Continued). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

850 

92 942 

947 

952 

957 

962 

967 

973 

978 

983 

988 



851 

993 

998 

*003 

*008 

*013 

*018 

*024 

*029 

*034 

*039 



852 

93 044 

049 

054 

059 

064 

069 

075 

080 

085 

090 



853 

095 

100 

105 

110 

115 

120 

125 

131 

136 

141 



854 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 



855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 



856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 


6 

857 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

l 

0.6 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

2 

1.2 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

3 

1.8 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

4 

2.4 

861 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

6 

3 6 

862 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

7 

4 2 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

8 

4 8 

864 

651 

656 

661 

666 

671 

676 

682 

687 

892 

897 

9 

5 4 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 



866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 



867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 



868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 



869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 



870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 



871 

94 002 

007 

012 

017 

022 

027 

032 

037 

042 

047 


5 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

i 

0.5 

873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

2 

1.0 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

3 

1.5 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

4 

2.0 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

5 

2.5 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

6 

3.0 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

7 

3.5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

8 

9 

4.0 

4.5 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 


881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 



882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 



883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 



884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 



885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 



886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 


4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

i 

0.4 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

2 

0.8 

889 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

3 

1.2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

4 

5 

1.6 

2 0 

891 

988 

993 

998 

*002 

*007 

*012 

*017 *022 

*027 

*032 

6 

2.4 

892 

95 036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

7 

2.8 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

8 

3.2 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

9 

3.6 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 



896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 



897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 



898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 



899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 



900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 


























































50 


MATHEMATICS 


Table— ( Continued). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P . 

900 

95 424 

429 

434 

4391 

444 

448 

453 

458 

463 

468 



901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 



902 

521 

525 

530 

535 

540 

545 

550 

554 

559 i 

564 



903 

569 

574 

578 

583 

588 

593 

598 1 

602 

607 

612 



904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 



905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 



906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 



907 

761 

766 

770 

775 

780 

785 

789 

794 ! 

799 

804 



908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 



909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 



910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 



911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 


5 

912 

999 

*004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 

1 

0.5 

913 

96 047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

2 

1.0 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

3 

1.5 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

4 

2.0 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

2.5 

' 917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

6 

3.0 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

7 

3.5 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

8 

9 

4.0 

4.5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 


921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 



922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 



923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 



924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 



925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 



926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 



927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 



928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 



929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 



930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 



931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 


4 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

i 

0.4 

933 

988 

993 

997 

*002 

*007 

*011 

*016 

*021 

*025 

*030 

2 

0.8 

934 

97 035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

3 

1.2 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

4 

1.6 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

2.0 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

6 

2.4 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

7 

2.8 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

8 

3.2 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

9 

3.6 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 



942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 



943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 



944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 



945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 



946 

589 

594 

598 

603 

i 607 

612 

617 

621 

626 

630 



947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 



948 

681 

685 

690 

! 695 

' 699 

704 

708 

713 

717 

722 



949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 



950 

772 

777 

782 

786 

| 791 

795 

800 

804 

809 

813 



N. 

L.O 

1 

2 

! 3 

4 

5 

6 

P * 

/ 

1 8 

9 

F 

\ p. 






























































MATH EM A TICS 


51 


Table —( Continued). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

p. 

950 

97 772 

777 

782 

786 

791 

795 

800 

804 

809 

813 



951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 



952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 



953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 



954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 



955 

98 000 

005 

009 

014 

019 

023 

028 

032 

037 

041 



956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 



957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 



958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 



959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 



960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 



961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 


5 

962 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

1 

0.5 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

2 

1.0 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

3 

1.5 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

4 

2.0 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

5 

2.5 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

6 

3.0 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

7 

* 5.0 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

8 

9 

4.0 

4.5 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 



971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 



972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 



973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 



974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 



975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 



976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 



977 

989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 



978 

99 034 

038 

043 

047 

052 

056 

061 

065 

069 

074 



979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 



980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 



981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 


4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

i 

0 . 4 * 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

2 

0.8 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

3 

1.2 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

4 

1.6 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

5 

2.0 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

6 

2.4 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

7 

2.8 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

8 

9 

3.2 

3 8 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 



991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 



992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 



993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 



994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 



995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 



996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 



997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 



998 

913 

917 

922 

926 

930 

935 

939 

944 

918 

952 



999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 



!000 

00 000 

004 

009 

013 

017 

022 

026 

030 

035 

039 



N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

_ 

8 

9 

p. p. 















































































52 


MA THEM A TICS 


GEOMETRY 

1. The sum of all the angles formed on one side of a straight 
line equals two right angles, or 180°. 

2. The sum of all the angles formed around a point equals 
four right angles, or 360°. 

3. When two straight lines intersect each other, the oppo¬ 
site or vertical angles are equal. 

4. If two angles have their sides parallel, they are equal. 

5. If two triangles have two sides and the included angle 
of the one equal to two sides and the included angle of the 
other, they are equal in all their parts. 

6. If two triangles have two angles and the included side 
of the one equal to two angles and the included side of the other, 
they are equal in all their parts. 

7. In any triangle, the greater side is opposite the greater 
angle, and the greater angle is opposite the greater side. 

8. The sum of the lengths of any two sides of a triangle 
is greater than the length of the third side. 

9. In an isosceles triangle, the angles opposite the equal 
sides are equal. 

10. In any triangle, the sum of the three angles is equal to 
two right angles, or 180°. 

11. If two angles of a triangle are given, the third may be 
found by subtracting their sum from two right angles, or 180°. 

12. A triangle must have at least two acute angles, and can 
have but one obtuse or one right angle. 

13. In any triangle, a perpendicular let fall from the apex 
to the base is shorter than either of the two other sides. 

14. In any parallelogram, the opposite sides and angles are 
equal each to each. 

15. The diagonals divide any parallelogram into two equal 
triangles. 

16. The diagonals of a parallelogram bisect each other; that 
is, they divide each other into equal parts. 

17. If the sides of a polygon are produced in the same direc¬ 
tion, the sum of the exterior angles will equal four right angles. 

18. The sum of the interior angles of a polygon is equal 
to twice as many right angles as the polygon has sides, less 


MA THEM AT ICS 


53 


four right angles. For example, the sum of the interior angles 
of a quadrilateral is (2X4)—4 = 4 right angles; the sum of the 
interior angles of a pentagon is (2X5)—4 = 6 right angles; 
the sum of the interior angles of a hexagon is (2 X 6) — 4 = 8 right 
angles. 

19. In equiangular polygons, each interior angle equals 
the sum divided by the number of sides. 

20. The square described on the hypotenuse of a right- 
angled triangle is equal to the sum of the squares described on 
the other two sides. Thus, in a right-angled triangle whose 
base is 20 ft. and altitude 10 ft., the square of the hypotenuse 
equals the square of 20 + the square of 10, or 500; then the 
hypotenuse equals the square root of 500, or 22.3607 ft. 

21. Having the hypotenuse and one side of a right-angled 
triangle, the other side may be found by subtracting from the 
square of the hypotenuse the square of the other known side; 
the remainder will be the square of the required side. 

22. Triangles that have an angle in each equal, are to each 
other as the product of the sides including those equal angles. 

23. Similar triangles are to each other as the squares of 
their corresponding sides. 

24. The perimeters of similar polygons are to each other 
as any two corresponding sides, and their areas are to each 
other as the squares of those sides. 

25. The diameter of a circle is greater than any chord. 

26. Any radius that is perpendicular to a chord, bisects 
the chord and the arc subtended by the chord. 

27. Through three points not in the same line, a circum¬ 
ference may be made to pass. For example, draw two lines 
connecting the three points and erect perpendiculars from the 
centers of each of these two lines; the point of intersection of 
the perpendiculars will be the center of the circle. 

28. The circumferences of circles are to each other as their 
radii, and their areas are to each other as the squares of their 
radii. 

Example 1.—If the circumference of a circle is 62.83 in. 
and its radius is 10 in., what is the circumference of a circle 
whose radius is 15 in.? 


54 


MATHEMATICS 


Solution. —Applying the principle just given, the circum¬ 
ference is 


10 : 15 = 62.83 : 94.245 in. 


Example 2.—If a circle 6 in. in diameter has an area of 
28.274 sq. in., what is the area of a circle 12 in. in diameter? 
Solution. —Applying the principle just given, the area is 


3 2 : 6 2 = 28.274 : 113.096 sq. in. 


MENSURATION 


In the following formulas, the letters have the meanings 
here given, unless otherwise stated. 

D = larger diameter; 
d = smaller diameter; 

R = radius corresponding to D; 
r — radius corresponding to d\ 
p — perimeter of circumference; 

C = area of convex surface, that is the area of flat surface that 
can be rolled into the shape shown; 

5 = area of entire surface ends = C+area of the end or ends; 

A = area of plane figure; 

7 r = 3.1416, nearly, that is the ratio of any circumference to 
its diameter; 

V = volume of solid. 

The other letters used will be found on the cuts. 


CIRCLE 


p = 7r d = 3.UlGd 
p = 2wr = G.2832r 


p = 2 VnA = 3.5449 Va 


2A 4 A 




P P 


= .1592£ 


2tt 6.2832 





MATHEMATICS 


55 




5642 Ya 


7T(i 2 

A = — = .7854d 2 
4 

A =7rr 2 = 3.1416i'2 
pd 

A = — = — 

2 4 


TRIANGLES 

D = £+C £ +S-fC = 

B = D — C E'+B+C = 
E'= E B' = B 

The above letters refer to angles. 
For a right-angled triangle, c being the 



180° 

180° 


hypotenuse. 


c= Va 2 +&2 

a = Vc 



■6 2 










v c <y 




I. X. 



















fc= Vc 2 —a 2 

c = length of side opposite an acute angle 
of an oblique-angled triangle. 

c = Va 2 +5 2 -26c 
/* = Va 2 — e 2 

length of side opposite an obtuse angle 
of an oblique-angled triangle. 



c= ^Ja 2 +b 2 +2be 
h = ^a 2 — e 2 

For a triangle inscribed in a 
angled triangle, c 


semicircle; i. 
: b = a : h 



right- 



ab ce 
c a 

a : b+e=e : a=h:c 


For any triangle, 


bh 

A = — = \bh 
2 


-hM 


a 2 +b 2 — c 2 \ 2 

2b I 































56 


MATHEMATICS 


RECTANGLE AND PARALLELOGRAM 

A =ab 

TRAPEZOID 

A = %h(a+b) 

TRAPEZIUM 

Divide into two triangles and a trapezoid. 

A = \bh'+ha{h'+h) + \ch 
or, A = h[bh'+ch+a(h'+h )] 

Or, divide into two triangles by drawing 
a diagonal. Consider the diagonal as the 
base of both triangles, call its length l; 
call the altitudes of the triangles hi and hi; then 

A = \l(hi+hz) 



b— a — 



b 




ELLIPSE 


D2 + J2 (D-d) 2 

p* = -rr\ - 

\ 2 8.8 

A = -Dd = ,7854Dd 
4 

SECTOR 

A = \lr 
irr^E 

A =-= .008727r 2 £ 

360 

l = length of arc 



SEGMENT 


A = 
A = 


% [Ir-c (r — h)] 
icr-E c 


360 


--(r-k) 


■jrrE 

1 = -=.0175 rE 

180 

180Z l 

E = -= 57.2956 - 

irr r 



* The perimeter of an ellipse cannot be exactly determined 
v/ithout a very elaborate calculation, and this formula is 
merely an approximation giving fairly close results. 















MA THEM A TICS 

RING 

A = -{D 2 —d 2 ) 

4 

CHORD 

c — length of chord 

_C 2 + 4h 2 «2 

8 /i_ ~_2h 

c = 2^2hr-h 2 
8 e — c 

1 = — : —, approximately 


'Ca — 



CONE 

C = \-ndl = 7 j-fl 

S = Trrl-\-Trr 2 = irr '\lr 2 -\-h 2 -\-irr 2 
ird 2 h .7854 d 2 h p 2 h 

4 X 3~ 3 _ 12tt 


57 



REGULAR POLYGONS 

Divide the polygon into equal triangles and find the sum of 
the partial areas. Otherwise, square the length of one side 
and multiply by proper number from the following table: 



Name 

No. Sides Multiplii 

Triangle.. 

. 3 

.433 

Square. .. 

. 4 

1.000 

Pentagon 

. 5 

1.720 

Hexagon. 

. 6 

2.598 

Heptagon 

. 7 

3.634 

Octagon.. 

. 8 

4.828 

Nonagon. 

. ... 9 

6.182 

Decagon. 

.10 

7.694 



IRREGULAR AREAS 

Divide the area into trapezoids, triangles, parts 
of circles, etc., and find the sum of the partial areas. 
If the figure is very irregular, the approximate area 
may be found as follows: Divide the figure into 
trapezoids by equidistant parallel lines b, c, d, etc. The lengths 






























58 


MA THEM AT ICS 


of these lines being measured, then, calling a the first and n 
the last length, and y the width of strips, 

( a-f-w 

—-—f-& + c-|-etc. + m 


PLANE TRIGONOMETRY 

DEFINITIONS 

Plane trigonometry treats of the solution of plane triangles. 
Every triangle consists of six parts, three sides and three angles. 
These parts are so related that when three are given, one being 
a side, the other parts may be found. 

An angle is measured by the arc included between its sides, 
the center of the circumference being at the vertex of the angle. 

For measuring angles, the circumference is divided into 360 
equal parts, called degrees ; each degree is divided into 60 equal 

parts called minutes. 

A quadrant is one-fourth the circum¬ 
ference of a circle, or 90°. 

The complement of an arc is 90° 
minus the arc; in Fig. 1, D C is the 
complement of BC, and the angle DOC 
is the complement of B O C. 

The supplement of an arc is 180° 
minus the arc; in Fig. 1, A E is the 
supplement of the arc B D E, and the 
angle A 0 E is the supplement of the angle B O E. 

In trigonometry, instead of comparing the angles of triangles 
or the arcs that measure them, the trigonometric functions 
known as the sine, cosine, tangent, cotangent, secant, and 
cosecant are compared. 

The sine, or sin, of an arc is the perpendicular let fall from 
one extremity of the arc on the diameter that passes through 
the other extremity; in Fig. 2, C D is the sine of the arc A C. 

The cosine, or cos, of an arc is the sine of its complement; or 
it is the distance from the foot of the sine to the center of the 
circle; in Fig. 2, C E or 0 D equals the cosine of arc A C. 



Fig. 1 





MATHEMATICS 


59 


The tangent, or tan, of an arc is a line perpendicular to the 
radius at one extremity of an arc and limited by a line passing 
through the center of the circle and the other extremity; in 
Fig. 2, A T is the tangent of A C. 

The cotangent, or cot, of an arc is equal to the tangent of the 
complement of the arc; in Fig. 2, B T' is the cotangent of A C. 

The secant, or sec, of 
an arc is a line drawn 
from the center of the 
circle through one ex¬ 
tremity of the arc, and 
limited by a tangent at 
the other extremity; in 
Fig. 2, 0 T is the secant 
of A C. 

The cosecant, or cosec, 
of an arc is the secant 
of the complement of 
the arc; in Fig. 2, the line 0 T' is the cosecant of the arc A C. 

The versed sine of an arc is that part of the diameter included 
between the extremity of the arc and the foot of the sine; in 
Fig. 2, DA is the versed sine of A C. 

The coversed sine is the versed sine of the complement of the 
arc; in Fig. 2, B E is the coversed sine of A C. 

From the above definitions, we derive the following simple 
principles: 

1 . The sine of an arc equals the sine of its supplement, and 
the cosine of an arc equals the cosine of its supplement. 

2. The tangent of an arc equals the tangent of its supple¬ 
ment, and the cotangent of an arc equals the cotangent of its 
supplement. 

3. The secant of an arc equals the secant of its supplement, 
and the cosecant equals the cosecant of its supplement. Thus, 


sin 

70° = sin 

O 

O 

r-H 

T-H 

cos 

70° = cos 

110 ° 

tan 

70° = tan 

110 ° 

cot 

70° = cot 

110 ° 

sec 

70° = sec 

110 ° 

cosec 

70° = cosec 

110 ° 


Thus, to find the sin 120° 30', look for the sin 180—120° 30'. 
or 59° 30', etc. 












60 MATHEMATICS 

In the right-angled triangle, xyz, Fig. 3, the following relations 
hold: 



The functions of the sum and difference of two angles are: 
sin (A +2?) = sin A cos B+cos A sin B 
cos (A + B) = cos A cos B — sin A sin B 
sin (A — B) = sin A cos B — cos A sin B 
cos (A —B) = cos A cos B-f sin A sin B 
Natural sines, tangents, etc. are calculated for a circle having 
a radius of unity, and logarithmic sines, tangents, etc. are cal¬ 
culated for a circle whose radius is 10,000,000,000. 


EXAMPLES IN SOLUTION OF TRIANGLES 

To Determine Height of Vertical Object Standing on Hori¬ 
zontal Plane.—Measure from the foot of the object any con¬ 
venient horizontal distance A B, Fig. 1; at the point A, take 
the angle of elevation B A C. Then, as B is known to be a 
right angle, two angles and the included side of a triangle are 
known. Assuming that the line A B is 300 ft. and the angle 
BAC = 40°, the angle C= 180°-(90°+40°) = 50°. Then, 
sinC: A B = sin A : BC, or, .766044: 300 = .642788: ( ) or, 251.73 
-f-ft. Or, by logarithms: 

Log 300= 2.4 7 7 1 2 1 
Log sin 40°= 9.808 0 67 

12.2 8 5 1 8 8 
Log sin 50°= 9.884 2 54 

2.4 0 0 9 3 4 or log of 251.73+ft. 

Hence, BC = 251.73+ ft. 

To Find Distance of Vertical Object Whose Height is Known. 

At a point A, Fig. 2, take the angle of elevation to the top of the 
object. Knowing that the angle B is a right angle, the angles 
B and A and the side B C of a triangle are known. Assuming 






MATHEMATICS 


61 


that the side B C = 200 ft. and the angle A =30°, the triangle 
is: Angle A =30°, -6 = 90°, C=60°, and the side B C = 200 ft. 

c 

i 

I 

I 

I 


Fig. 1 

Then, sin A : B C = sin C : A B, or .5 : 200 = .866025:, or 
346.41 ft. By logarithms: 

Log 200= 2.3 0 1 030 
Log sin 60°= 9.9 3 7 53 1 

1 2.2 3 8 5 6 1 
Log sin 30°= 9.6 9 89 7 0 

2.5 3 9 5 9 1 or log of 346.41 ft. 

To Find Distance of Inaccessible Object.—Measure a hori¬ 
zontal base line A B, Fig. 3, and take the angles formed by the 
lines B A C and ABC; this 
gives the two angles and the 
included side. Assuming the 
angle A = 60°, the angle B = 50°. 
and the side A 13 = 500 ft., 
angle C = 180° - (60° + 50°) 

= 70°. Then, 
sin 70° : A B = sin A : B C, 
and 

sin 70° : A B = sin B : A C; 
or, .939693 : 500 = .866025 : B C, or 460.8+ , 

and .939693 : 500 = .766044 : A C, or 407.6+. 

By logarithms: 

Log 500 = 2.6 9 8 9 7 0 
Log sin 60° = 9.9 3 7 5 3 1 

1 2.6 3 6 5 0 1 
Log sin 70°= 9.9 72 9 86 



Fig. 3 




2.6 6 3 5 1 5 = log of 460.8 + 




































62 


MA THEM A TICS 


Log 500= 2.69 8 9 70 

Log sin 50°= 9.8 8 4 2 5 4 
1 2.583 224 
Log sin 70°= 9.9 7 2 9 8 6 

2.6 1 0 2 3 8 = log of 407.6 + 


To Find Distance Between Two Objects Separated by an 
Impassable Barrier.—Select any convenient station, as C, 

Fig. 4, measure the lines C A and 
H C B, and the angle included between 
these sides; this gives the two sides 
and the included angle. Assuming 
angle C = 60°, the side C A = 600 ft., 
and side CB = 500 ft., 

CA+CB :C A-CB = tan 

A+B B-A 

- :tan - 

2 2 

180°-60° 

--, or 60° 

2 

B-A 



C 

Fig. 4 


Then, 


A+B 


Then, 


1,100 : 100 = tan 60° : tan' 


or, 1,100 : 100 =1.732050 : .157459, or tangent of-» 

2 

or 8° 57'. 

Then, 60°-f-8° 57' = 68°57\ or angle B, and 60° —8° 57' = 51° 
03', or angle A. Having found the angles, find the third side 
by the method given in connection with Fig. 1. 

The foregoing formula, worked out by logarithms, is as fol¬ 
lows: Log 100= 2.0 0 00 0 0 

Log Tan 60°= 1 0.23 85 6 1 

12.23 856 1 
Log 1,100= 3.04 1 3 93 

9.1 9 7 1 6 8 = log tan of ——— , or 8° 57' 

2 

Then, 60°+8° 57'= 68° 57', or angle B, and 60°-8° 57'= 51° 
03', or angle .4. 

Note. —The greater angle is always opposite the greater 
side. 














MATHEMATICS 


63 


To Find Height of Vertical Object Standing Upon Inclined 
Plane. —Measure any convenient distance D C, Fig. 5, on a 
line from the foot of the object, and, at the point D, measure 
the angles of elevation EDA and ED B to foot and top of 
tower. This gives two triangles, both of which may be solved 
by the method given in connection with Fig. 1, and the height 
above D of both the foot and top will be known. The difference 
between them is the height of the tower. 

To Find Height of Inaccessible Object Above a Horizontal 
Plane. — First Method. —Measure any convenient horizontal 
line A B, Fig. 6, directly toward the object, and take the angles 




of elevation at A and B. Assuming the line A 5 = 1,200 ft., 
the angle A =25°, and the angle DBC = 40°; then angle ABC 
= 180° —40°= 140°. Then, having the side B C, and the angle 
D B C = 40°, and the angle B D C = 90°, we find the side C D 
by the same method given in connection with Fig. 1. 

Second Method. —If it is not convenient to measure a horizon¬ 
tal base line toward the object, measure any line A B, Fig. 7, 
and also measure the horizontal angles B A D, A B D, and the 
angle of elevation D B C. Then, by means of the two triangles 
A B D and C B D, the height C D can be found. With the 
line A B and the angles BAD and A B D known, two angles 
and the included side are known. The third angle is then readily 
found and the side B D can be found. In the triangle B DC 
the angle B is known; by measurement, D = 90°, and the side 
B D is known. Then, the side C D, or the vertical height, can 
be found by the method given in connection with Fig. 1. 























64 


MATHEMATICS 


To Find Distance Between Two Inaccessible Objects When 
Points Can Be Found From Which Both Objects Can Be Seen. 

Wishing to know the horizontal distance between a tree and a 
house on the opposite side of a river, measure the line A B, 
Fig. 8, and, at point A , take the angles D A C, and DAB, and, 
at the point B, take the angles C B A and C B D. Assume the 
length of A 23 = 400 ft.; angle DA C = 56° 30'; angle DAB 
= 42° 24'; angle C B A =44° 36'; angle CBD = 68° 50'. In 
the triangle A B D, A 23 = 400 ft., angle D A 23 = 42° 24', angle 
A B D= (44° 36'+68° 50') = 113° 26', and angle A D B= 180° 



— (42° 24' +113° 26') = 24° 10'. Then, according to the method 
given in connection with Fig. 1, find the side D 23; this gives 
three angles and two sides of the triangle A D 23. The third 
side AD is found by the same method. In the triangle ABC 
the angles ABC and 23 A C, and the distance A B are known, 
so that the side A C may be found. Then, in the triangle 
ADC, the sides A D and A C, and the angle D AC, are known 
and the side C D may be found by the method given in con¬ 
nection with Fig. 4. 

TABLE OF TRIGONOMETRIC FUNCTIONS 

The following table contains the natural sines, cosines, tan¬ 
gents, and cotangents of angles from 0° to 90°. Angles less 
than 45° are given in the first column and the names of the 
functions are given at the top; angles greater than 45° appear at 
the right-hand side of the page, and the names of the functions 
are given at the bottom. Thus, the second column contains 




MATHEMATICS 


65 


the sines of angles less than 45° and the cosines of angles 
greater than 45°; the sixth column contains the cotangents of 
angles less than 45° and the tangents of angles greater than 45°. 

To find the function of an angle less than 45°, look in the 
first column for the angle, and at the top of the page for the 
name of the function; to find a function of an angle greater 
than 45°, look in the column at the right of the page for the 
angle and at the bottom of the page for the name of the func¬ 
tion. The successive angles differ by an interval of 10'; they 
increase downwards in the left-hand column and upwards in the 
right-hand column. Thus, for angles less than 45° read down 
from top of page, and for angles greater than 45° read up from 
bottom of page. 

The columns headed d contain the differences between the 
successive functions. For example, the sine of 32° 10' is .5324 
and the sine of 32° 20' is .5348; the difference is .5348 —.5324 
= .0024, so that 24 is written on the third column, just opposite 
the space between .5324 and .5348. In like manner the differ¬ 
ences between the successive tabular values of the tangents are 
given in the fifth column, those between the cotangents in the 
seventh column, and those for the cosines in the ninth column. 
These differences in the functions correspond to a difference of 
10' in the angle; thus, when the angle 32° 10' is increased by 
10', that is, to 32° 20', the increase of the sine is .0024, or, as 
given in the table, 24. In the tabular difference, no attention 
is paid to the decimal point, as the difference is merely the 
number obtained by subtracting the last two or three figures 
of the smaller function from those of the larger. 

These differences are used to obtain the sines, cosines, etc., 
of angles not given in the table. For example, suppose that the 
tangent of 27° 34' is required. Looking in the table, it is 
found that the tangent of 27° 34' is .5206, and (column 5) the 
difference for 10' is 37, or when written in full .0037. As the 
difference for 1' is .0037-M0 = .00037, the difference for 4' is 
.00037X4 = .00148. Adding this difference to the value of the 
tan 27° 30' gives 

tan 27° 30'= .5206 
difference for 4'= .00148 

tan 27° 34'= .52208 or .5221, to 4 places 



66 


MA THEM A TICS 


Because only 4 decimal places are retained, the 8 in the fifth 
place is dropped and the fourth figure is increased by 1, as 8 is 
greater than 5. 

Column of Proportional Parts.—To avoid multiplication, 
the column of proportional parts, headed P. P., is used. At 
the head of each table in this column is the difference for 10', 
and below are the differences for any intermediate number 
of minutes from 1' to 9'; these differences are written as whole 
numbers and decimal parts of same. In the example the 
difference for 10' was 37; looking in the table with 37 at the 
head, the difference opposite 4 is 14.8; that opposite 7 is 25.9; 
and so on. For want of space, the differences for the cotangents 
for angles less than 45° (or the tangents of angles greater than 
45°) have been omitted from the tables of proportional parts. 
The use of these functions should be avoided, if possible, for 
the differences change so rapidly that computation is likely to 
be inexact. 

The method to be employed when dealing with these func¬ 
tions is shown in the following example, in which the tangent 
of 76° 34' is required. Because this angle is greater than 45° 
it is found in the column at the right, which is to be read 
upwards. Opposite 76° 30', in the sixth column, is found the 
number 4.1653; and corresponding to it, in the seventh column, 
is found the difference 540. As 540 is the difference for 10', the 
difference for 4' is 540X^ = 216. Putting this number in its 
true form and adding gives 

tan 76° 30'= 4.1653 
difference for 4'= .0216 

tan 76° 34'= 4.1869 

Angles Containing Seconds.—When the angle contains a 
certain number of seconds, divide the number by 6, and take the 
whole number nearest to the quotient. Find this number in the 
table of proportional parts (under the proper difference), and 
take out the number that is opposite to it. Shift the decimal 
point one place to the left, and then add it to the partial function 
already found. The following examples represent the methods 
of using the tables of proportional parts for the different 
functions: For example, find the sine of 34° 26' 44". 






MATHEMATICS 


67 


sin 34° 20' =.5640 Difference for 10'= 24, or .0024 

difference for 6'= .00144 

difference for 44" = .00017 - 4 S 4 - = 7|. In the P.P. table, the 

sin 34° 26' 44" = .5656 number under 24 and opposite 

7 is found to be 16.8. Shift¬ 
ing the decimal point one place 
to the left gives 1.68, or, 1.7, 
which when put in its decimal 
form is .00017. 

The tangent is found in the same way as the sine. 

As the angle increases the value of the cosine decreases, there¬ 
fore, to find the cosine of an angle, instead of adding the values 
corresponding to 6' and 44" to the function already found, sub¬ 
tract them from it. Thus, find the cosine of 34° 26' 44". 

cos 34° 20'= .8258 Difference for 10' = 17, or .0017. 

difference for 6' = .00102 

difference lor 44" = .00012 The number under the 17 and 

total difference = .00114 opposite the 7, in the P. P. 

g 2 ^j table, is 11.9. Therefore take 
1.19, or, say, 1.2, which may 
be written .00012. 

Therefore, cos 34° 26'44" = .8258-.0011 = .8247. Only 4 
decimal places are kept; therefore, the figure of the difference 
following the decimal point is dropped before subtracting. 

The cotangent is found in the same manner. 

To show the method when the angle is greater than 45°, 
suppose that it is required to find the sine of 68° 47' 22". In 
obtaining the difference, it must be remembered to choose the 
one between the sine of 68° 40' and the next angle above it, 
namely, 68° 50'. 

sin 68° 40'= .9315 Difference for 10' = 10, or .0010. 

difference for 7'= .0007 

difference for 19" = .00004 -^ = 3§, say 4. Under the 10 

sin 68° 47' 22"= 9322 and opposite the 4 is the 

number 4.0; shifting the deci¬ 
mal point, gives .4, or .00004. 

As usual, only 4 decimal places are kept. 

The tangent is found in a similar manner to the method just 
given. 






€8 


MAT HEM A TICS 


As before, the cosine decreases as the angle increases; there¬ 
fore, to find the cosine of 68° 47' 22", subtract the successive 
sine values corresponding to the increments in the angle. 

cos 68° 40'= .3638 Difference for 10'= 27, or, .0027. 

difference for 7'= .00189 

difference for 22" = .00011 Under the 27 and opposite the 

total difference = .0020 4 is the number 10.8; there- 

■gglg' fore, take 1.08 in this case, 
or, say, 1.1, which may be 
written .00011. 

Therefore, cos 68° 47' 22" = .3638- .002 = .3618. 

The cotangent is found in the same way. 

Applying Proportional Differences.—In finding the functions 

of an angle, the only difficulty likely to be encountered is to 
determine whether the difference obtained from the table of 
proportional parts is to be added or subtracted. This can be 
told by observing whether the function is increasing or decreas¬ 
ing as the angle increases. For example, take the angle 21°; its 
sine is .3584, and the following sines, reading downwards, 
are .3611, .3638, etc. Therefore, the sine of say 21° 6' is greater 
than that of 21° and the difference for 6' must be added. On 
the other hand, the cosine of 21° is .9336, and the following 
cosines, reading downwards, are .9325, .9315, etc.; that is, as 
the angle grows larger the cosine decreases. The cosine of an 
angle between 21° and 21° 10', say 21° 6', must therefore lie 
between .9325 and .9315; that is, it must be smaller than .9325, 
which shows that the difference for 6' must be subtracted from 
the cosine of 21°. 

Finding the Angle.—Find the angle whose sine is .4943. 
The operation in this case may be arranged as follows: 

.4943 Difference for 10'= 26, or .0026. 
. 4924 = sin 29° 30'. 

1st remainder .0019 

.00182 = difference for 7'. 

2 d remainder .00008 

As .78 is the difference for .3' or 18", .4943 = sin. 29° 37' 18". 

Looking down the second column, the sine next smaller than 
.4943 is found to be .4924, and the difference for 10' to be 26. 







MATHEMATICS 


69 


The angle corresponding to .4924 is 29° 30'. Subtracting 
the .4924 from 4943, the first remainder is 19. Looking in the 
table of proportional parts, the part next lower than this differ¬ 
ence is 18.2, opposite which is 7'. Subtracting this difference 
from the remainder gives .8, and, looking in the table, it is 
found that 7.8 with its decimal point moved one place to the 
left is nearest to the second difference. This is the difference 
for .3'or 18". Hence, the angle is 29°30' + 7' + 18" = 29 o 37' 18". 

Find the angle whose tangent is .8824. 

.8824 Difference for 10'= 51, or .0051. 
.8796 =tan 41° 20'. 

1st remainder .0028 

.00255 = difference for 5'. 

2d remainder .00025 

As 2.55 is the difference for .5' or 30", .8824 = tan 41° 25' 30". 

In the examples just given, the minutes and seconds corre¬ 
sponding to the 1st and 2d remainders are added to the angle 
taken from the table. Thus, in the first example, an inspection 
of the table shows that the angle increases as the sine increases; 
hence, the angle whose sine is .4943 must be greater than 29° 30', 
whose sine is .4924. For this reason the correction must be 
added to 29° 30'. The same reasoning applies to the second 
example. 

Find the angle whose cosine is .7742. 

.7742 Difference for 10'= 18, or .0018. 
.7735 = cos 39° 20'. 

st remainder .0007 

.00054 = difference for 3'. 

2d remainder .00016 

As 1.62 is the difference for .9' or 54", 39° 20' —3' 54" = 
39° 16' 6", which is the angle whose cosine is .7742. 

Looking down the column, headed cos, the next smaller 
cosine is .7735, to which corresponds the angle 39° 20'. The 
difference for 10' is 18. Subtracting, the remainder is 7, and 
the next lower number in the table of proportional parts is 
5.4, which is the difference for 3'. Subtracting this from first 
remainder, the second remainder is 1.6, which is nearest 16.2 
of table of proportional parts, if the decimal point of the latter 






70 


MA THEM A TICS 


is moved to the left one place. As 16.2 corresponds to a differ¬ 
ence of 9', 1.62 corresponds to a difference of .9', or 54". Hence, 
the correction for the angle 39° 20' is 3' 54". From the table, 
it appears that, as the cosine increases, the angle grows smaller; 
therefore, the angle whose cosine is .7742 must be smaller than 
the angle whose cosine is .7735, and the correction for the angle 
must be subtracted. 

Find the angle whose cotangent is .9847. 

.9847 Difference for 10' = 57, or .0057. 

.9827 =cos 45° 30'. 

1st remainder .0020 

.00171 = difference for 3'. 

2d remainder .00029 

As 2.85 is the difference for .5' or 30", 45° 30' —3'30" = 
45° 26' 30", the angle whose cotangent is .9847. 

In finding the angle corresponding to a function, as in 
the foregoing examples, the angles obtained may vary from the 
true angle by 2 or 3 sec.; in order to obtain the number of 
seconds accurately, the functions should contain 6 or 7 decimal 
places. 


i 




o 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 


Tan. d. Cot. d. Cos. d. 


>.0000 


>.0029 

>.0058 

>.0087 

>.0116 

>.0145 


>.0173 


>.0204 

>.0233 

>.0262 

>.0291 

>.0320 


>.0349 


>.0378 

>.0407 

>.0437 

>.0466 

>.0493 


>.0524 

>.0553 

>.0582 

>.0612 

>.0641 

>.0670 


>.0699 


>.0729 

>.0758 

>.0787 

>.0816 

t.0846 


1.0875 


1.0904 

1.0934 

'.0963 

'.0992 

'.1022 


.1051 


.1080 

.1110 

.1139 

.1169 

.1198 


.1228 


.1257 

.1287 

.1317 

.1346 

.1376 


.1405 


.1435 

.1463 

.1495 

.1524 

.1554 


.1584 


29 

29 

29 

29 

29 

30 

29 

29 

29 

29 

29 

29 

29 

29 

30 
29 
29 

29 

29 

29 

30 
29 
29 

29 

30 
29 
29 

29 

30 

29 

29 

30 
29 

29 

30 

29 

29 

30 

29 

30 

29 

30 

29 

30 
30 

29 

30 

29 

30 
30 
30 

29 

30 

30 


inlinit. 


343.7737 

171.8854 

114.5887 

85.9398 

68.7501 


57.2900 


49.1039 

42.9641 

38.1885 

34.3678 

31.2416 


28.6363 


26.4316 

24.5418 

22.9038 

21.4704 

20.2056 

19.0811 


18.0750 

17.1693 

16.3499 

15.6048 

14.9244 

14.3007 

13.7267 

13.1969 

12.7062 

12.2505 

11.8262 


11.430 1 

11.0594 

10.7119 

10.3854 

10.0780 

9.7882 

9.5144 


9.2553 

9.0098 

8.7769 

8.5555 

8.3450 


8.1443 


7.9530 

7.7704 

7.5958 

7.4287 

7.2687 


7.1154 

6.9682 

6.8269 

6.6912 

6.5606 

6.4348 

6.3138 


vOt. d. Tan. d. Sin. Id. 


81861 

61398 

47756 

38207 

31262 

26053 

22047 

18898 

16380 

14334 

12648 

11245 

10061 

905' 

8194 

7451 

6804 

6237 

5740 

5298 

4907 

4557 

4243 

3961 

3707 

3475 

3265 

3074 

2898 

2738 

2591 

2455 

2329 

2214 

2105 

2007 

1913 

1826 

1746 

1671 

1600 

1533 

1472 

1413 

1357 

1306 

1258 

1210 


1.0000 


1.0000 

1.0000 

1.0000 

0.9999 

0.9999 

0.9998 

0.9998 

0.9997 

0.9997 

0.9996 

0.9995 

0.9994 

0.9993 

0.9992 

0.9990 

0.9989 

0.9988 

0.9986 

0.9985 

0.9983 

0.9981 

0.9980 

0.9978 

0.9976 

0.9974 

0.9971 

0.9969 

0.9967 

0.9964 

0.9962 

0.9959 

0.9957 

0.9954 

0.9951 

0.9948 

0.9945 

0.9942 

0.9939 

0.9936 

0.9932 

0.9929 

0.9925 

0.9922 

0.9918 

0.9914 

0.9911 

0.9907 

0.9903 

0.9899 

0.9894 

0.9890 

0.9886 

0.9881 


0.9877 


0 

90 

50 


40 


30 


20 


10 


0 

89 

50 


40 


30 


20 


10 


0 

88 

50 


40 


30 


20 


10 


0 

87 

50 


40 


30 


20 


10 


0 

86 

50 


40 


30 


20 


10 


0 

85 

50 


40 


30 


20 


10 


0 

84 

50 


40 


30 


20 


10 


0 

83 

50 


40 


30 


20 


10 


0 

82 

50 


40 


30 


20 


10, 


0 

81 

r 

o 


P. P. 


30 

3.0 

6.0 

9.0 

12.0 

15.0 

18.0 

21.0 

24.0 

27.0 


29 

2.9 

5.8 

8.7 

11.6 

14.5 

17.4 

20.3 

23.2 

26.1 


28 

2.8 

5.6 

8.4 

11.2 

14.0 

16.8 

19.6 

22.4 

^ j.2 


5 

0.5 

1.0 

1.5 

2.0 

2.5 
3.0 

3.5 
4.0 

4.5 


4 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.8 

3.2 

3.6 


P. P. 
















































































































o 

9 

10 

II 

12 

13 

14 

15 

16 

17 

18 


6.3138 


0.1673 

0.1703 

0.1733 

0.1763 


0.1793 

0.1823 

0.1853 

0.1883 

0.1911 


0.1944 

0J974 

0.2004 

0.2035 

0.2065 

0.2095 


0.2126 

(42156 
0.2186 
0.2217 
0.2247 
0.22781 

0.2339 31 
0.237013i 
0.24011 31 
0-2432 30 
0.2462 L 

0-2493 j 

0.252i; 31 
0.2555 31 
0.2586 31 
0.2617 31 
0.2648 


0.2679 


0.2711 

0.2742 

0.2773 

0.2805 

0.2836 

0.2867 

0.2899 

0.2931 

0.2962 

0.2994 

0.3026 

0.3057 


0.3089 
0.3121 
0.3153!on 
0.318 . 
0.3217 


0.3249 


6.1970 

6.0844 

5.9758 

5.8708 

5.7694 


Tan. d. Cot. d. Cos. d. 

0.1584j “ 

0.1614|qn 
0.1644 09 

o " 30 

30 
30 

30 
30 
30 

30 

31 

30 

30 

30 

31 
30 

30 

31 


5.6713 

5.5764 

5.4845 

5.3955 

5.3093 

5.2257 

5.1446 

5.0658 

4.9894 

4.9152 

4.8430 

4.7729 

4.7046 

4.6382 

4.5736 

4.5107 

4.4494 

4.3897 

4.3315 

4.2747 

4.2193 

4.1653 

4.1126 

4.0611 

4.0108 

3.9617 

3.9136 

3.8667 

3.8208 

3.7760 

3.7321 

3.6891 

3.6470 

3.6059 

3.5656 

3.5261 

3.4874 

3.4495 

3.4124 

3.3759 

3.3402 

3.3052 

3.2709 

3.2371 

3.2041 

3.1716 

3.1397 

3.1084 

3.0777 


981 

949 

919 

890 

862 

836 

811 

788 

764 

742 

722 

701 

683 

664 

646 

629 

613 

597 

582 

568 

554 

540 

527 

515 

503 

491 

481 

469 

459 

448 

439 

430 

421 

411 

403 

395 

387 

379 

371 

365 

357 

350 

343 

338 

330 

325 

319 

313 

307 


0.9877 

0.9872 

0.9868 

0.9863 

0.9858 

0.9853 

0.9848 

0.9843 

0.9838 

0.9833 

0.9827 

0.9822 

0.9816 

0.9811 

0.9805 

0.9799 

0.9793 

0.9787 

0.9781 

0.9775 

0.9769 

0.9763 

0.9757 

0.9750 

0.9744 

0.9737 

0.9730 

0.9724 

0.9717 

0.9710 

0.9703 

0.9696 

0.9689 

0.9681 

0.9674 

0.9667 

0.9659 

0.9652 

0.9644 

0.9636 

0.9628 

0.9621 

0.9613 

0.9605 

0.9596 

0.9588 

0.9580 

0.9572 

0.9563 

0.9555 

0.9546 

0.9537 

0.9528 

0.9520 

0.9511 

Sin. 


0 

81 

50 


40 


30 


20 


10 


0 

80 

50 


40 


30 


20 


10 


0 79 

50 


40 


30 


20 


10 


0 

78 

50 


40 


30 


20 


10 


0 

77 

50 


40 


30 


20 


10 


0 7C 

50 


40 


30 


20 


10 


0 

75 

50 


40 


30 


20 


10 


0 74 

50 


10 


30 


20 


10 


0 73 

50 


40 


30 


20 


10 


0 

72 

/ 

O 



P 

P. 



32 

31 

30 

l 

3.2 

3.1 

3.0 

2 

6.4 

6.2 

6.0 

3 

9.6 

9.3 

9.0 

4 

12 8 

12.4 

12.0 

5 

16.0 

15.5 

15.0 

6 

19.2 

18.6 

18.0 

7 

22.4 

21.7 

21.0 

8 

25.6 

24.8 

24.0 

9 

28.8 

27.9 

27.0 


29 

28 

27 

1 

2.9 

2.8 

2.7 

2 

5.8 

5.6 

5.4 

3 

8.7 

8.4 

8.1 

4 

11.6 

11.2 

10.8 

5 

14.5 

i 4.0 

13.5 

6 

17.4 

16.8 

16.2 

7 

20.3 

19.6 

18.9 

8 

23.2 

22.4 

21.6 

9 

26.1 

25.2 

24.3 



9 

8 

1 

0.9 

0.8 

2 

1.8 

1.6 

3 

2.7 

2.4 

4 

3.6 

3.2 

5 

4.5 

4.0 

6 

5.4 

4.8 

7 

6.3 

5.6 

8 

7.2 

6.4 

9 

8.1 

7.2 


7 

6 

1 

0.7 

0.6 

2 

1.4 

1.2 

3 

2.1 

1.8 

4 

2.8 

2.4 

5 

3.5 

3.0 

6 

4.2 

3.6 

7 

4.9 

4.2 

8 

5.6 

4.8 

9 

6.3 

5.4 


1 

2 

3 

4 

5 

6 

7 

8 
9 


5 4 

0.5 0.4 

1 . 00.8 

1.5 1.2 
2.0!1.6 
2.52.0 
3.0 2.4 

3.5 2.8 
4.0 3.2 
4.5[3.6 


P. P. 


72 









































































































o 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 


. Cot. 

3.0777 

3.0473 

3.0178 

2.9887 

2.9600 

2.9319 

2.9042 

2.8770 

2.8502 

2.8239 

2.7980 

2.7725 

2.747j 

2.7228 

2.6985 

2.6746 

2.6511 

2.6279 

2.6051 

2.5826 

2.5603 

2.5386 

2.5172 

2.4960 

2.4751 

2.4545 

2.4342 

2.4142 

2.3947 

2.3750 

2.3559 

2.3369 

2.3183 

2.2998 

2.2817 

2.2637 

2.2460 

2.2286 

2.2113 

2.1943 

2.1773 

2.1609 

2.1445 

2.1283 

2.1123 

2.0965 

2.0809 

2.0655 

2.0503 

2.0353 

2.0204 

2.0057 

1.9912 

1.9768 

1.9626 

Tan. 


3.3249 


3.3281 

3.3314 

3.3346 

3.3378 

3.3411 


33 


3.3443 32 

3.3476 loo 
3.35(38 oo 
3.3541 loo 
3.3574 loo 
1.3607 rf 
33 

33 
33 
33 
33 
33 

1.38391 34 

—-33 

L3872 34 
.3906 oo 
.3939 34 
.3973 33 


(.3640 

1.3673 

1.3706 

1.3739 

1.3772 

1.3895 


l 

.4006 


.4074 34 
.4108 34 
•4X42 
•4176 34 

^“35 

424 '34 


.4279 35 
.43I4I34 
.4348'35 
.4383 04 
.4417 


.4452 

.4487 

.4522 

.4557 

,459211 

.4628 

,4663 

4699 

4734 

4770 

4806 

4841 


4877 


4913 
49 0 
,4986 
.5022 
.5059 

'5095 


35 

35 

35 

35 

35 

36 

35 

36 

35 

36 
36 

35 

36 

36 

37 
36 

36 

37 

36 


302 

29 

291 

28 

281 

277 

27: 

268 

263 

259 

255 

250 

247 

243 

239 

235 

232 

228 

225 

221 

219 

214 

212 

209 

206 

203 

200 

197 

195 

191 

190 

186 

185 

181 

180 

177 

174 

173 

170 

168 

166 

164 

162 

160 

158 

156 

154 

152 

150 

149 

147 

145 

144 

142 


0.9511 


0.9502 

0.9492 

0.9483 

0.9474 

0.9465 


0.9455 


0.9446 

0.9436 

0.9426 

0.9417 

0.9407 

679397 


0.9387 

0.9377 

0.9367 

0.9356 

0.9346 

079336 


0.9325 

0.9315 

0.9304 

0.9293 

0.9283 

679272 

679261 

0.9250 

0.9239 

0.9228 

0.9216 


0.9205 

6.9194 

0.9182 

0.9171 

0.9159 

0.9147 


0.9135 


0.9124 

0.9112 

0.9100 

0.9088 

0.9075 

0.9063 


0.9051 

0.9038 

0.9026 

0.9013 

0.9001 

678988 


0.8975 

0.8962 

0.8949 

0.8936 

0.8923 

0.8910 


Sin. d. 


11 

10 

11 

11 

10 

11 

11 

11 

11 

11 

12 

11 

11 

12 

11 

12 

12 

12 

11 

12 

12 

12 

13 

12 

12 

13 

12 

13 

12 

13 

13 

13 

13 

13 

13 

13 


0 72 

50 
40 
30 
20 
10 

0 

50 
40 
30 
20 
10 

0 

50 
40 
30 
20 

10 


7 


70 


0 69 

50 
40 
30 
20 
10 

0 68 

50 
40 
30 
20 
10 

0 67 

50 

40 

30 

20 

10 

0 66 

50 

40 

30 

20 

10 

0 65 

50 

40 

30 

20 

10 

0 64 

50 

40 

30 

20 

10 

0 63 


p. r. 


37 

3.7 

7.4 

11.1 

14.8 

18.5 
22.2 

25.9 

29.6 
33.3 


36 

3.6 

7.2 

10.8 

14.4 

18.0 

21.6 

25.2 

28.8 


35 

3.5 

7.0 

10.5 
14.0 

17.5 

21.0 

24.5 
28.0 



34 

33 

1 

3.4 

3.3 

2 

6.8 

6.6 

3 

10.2 

9.9 

4 

13.6 

13.2 

5 

17.0 

16.5 

6 

20.4 

19.8 

7 

23.8 

23.1 

8 

27.2 

26.4 

9 

30.6 

29.7 


32.4 31.5 


32 

3.2 

6.4 

9.6 

12.8 

16.0 

19.2 
22.4 
25.6 
28.8 



28 

27 

26 

1 

2.8 

2.7 

2.6 

2 

5.6 

5.4 

5.2 

3 

8.4 

8.1 

7.8 

4 

11.2 

10.8 

10.4 

5 

14.0 

13.5 

13.0 

6 

16.8 

16.2 

15.6 

7 

19.6 

18.9 

18.2 

8 

22.4 

21.6 

20.8 

9 

25.2 j 24.3 

23.4 


13 

1.3 
2.6 
3.9 
5.2 
6.5 
7.8 
9.1 
10.4 
11-71 


12 

1.2 

2.4 

3.6 
4.8 
6.0 

7.2 

8.4 

9.6 

10.8 


II 11C 

1.1 1.0 


9 

0.9 

1.8 

2.7 

3.6 

4.5 


3.3i3.0 

4.4 4.0 

5.5 5.0 

6.6 6.0 5.4 

7.7 7.0 6.3 

8.8 8.0 7.2 
9.9,9.08.1 


P. P. 














































































































o 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 


Cot. 

1.9626 

1.9486 

1.9347 

1.9210 

1.9074 

1.8940 

1.8807 

1.8676 

1.8546 

1.8418 

1.8291 

1.8165 

1.8040 

1.7917 

1.7796 

1.7675 

1.7556 

1.7437 

1.7321 

1.7205 

1.7090 

1.6977 

1.6864 

1.6753 

1.6643 

1.6534 

1.6426 

1.6319 

1.6212 

1.6107 

1.6003 

1.5900 

1.5798 

1.5697 

1.5597 

1.5497 

1.5399 

1.5301 

1.5204 

1.5108 

1.5013 

1.4919 

1.4826 

1.4733 

1.4641 

1.4550 

1.4460 

1.4370 

1.4281 

1.4193 

1.4106 

1.4019 

1.3934 

1.3848 

1.3764 


Cos. 

0.8910 

0.8897 

0.8884 

0.8870 

0.8857 

0.8843 

0.8829 

0.8816 

0.8802 

0.8788 

0.8774 

0.8760 

0.8746 

0.8732 

0.8718 

0.8704 

0.8689 

0.8675 

0.8660 

0.8646 

0.8631 

0.8616 

0.8601 

0.8587 

0.8572 

0.8557 

0.8542 

0.8526 

0.8511 

0.8496 

0.8480 

0.8465 

0.8450 

0.8434 

0.8418 

0.8403 

0.8387 

0.8371 

0.8353 

0.8339 

0.8323 

0.8307 

0.8290 

0.8274 

0.8258 

0.8241 

0.8223 

0.8208 

0.8192 

0.8175 

0.8158 

0.8141 

0.8124 

0.8107 

0.8090 

Sin. 



P 

P. 



44 

43 

1 1 

1 

4.4 

4.3 

4.2 

2 

8.8 

8.6 

8.4 

3 

13.2 

12.9 

12.6 

4 

17.6 

17.2 

16.8 

5 

22.0 

21.5 

21-.0 

6 

26.4 

25.8 

25.2 

7 

30.8 

30.1 

29.4 

8 

35.2 

34.4 

33.6 

9 

39.6 

38.7 

37.8 


41 

40 

39 

1 

4.1 

4.0 

3.9 

2 

8.2 

8.0 

7.8 

3 

12.3 

12.0 

11.7 

4 

16.4 

16.0 

15.6 

5 

20.5 

20.0 

19.5 

6 

24.6 

24.0 

23.4 

7 

28.7 

28.0 

27.3 

8 

32.8 

32.0 

31.2 

9 

36.9 

36.0 

35.1 


0.5095 

_ 97 

0.5132!oi 
0.5169 97 
0.5206 
0.5243;07 
0.52801 

37 


0.5317 


37 


QQ 

0.5392 00 
0.5430 07 
0.5467 0 = 
0.5505 38 

0.5543 38 
_i_ 30 

0.5581 as 
0.5619 3c ) 
0.5658 3g 
0.5696 aq 
0.5735 

39 

38 

39 

39 

40 

39 

40 


0.5774 

0.5812 

0.5851 

0.5890 

0.5930 

0.5969 


0.6048 

0.6088 

0.6128 

0.6168 

0.6208 


0.6249 


0.6289 

0.6330 

0.6371 

0.6412 

0.6453 


0.6494 


°- 6009 39 
40 
40 
40 

40 

41 

40 

41 
41 
41 
41 

41 

42 

41 

42 
42 
42 

42 

42 

43 
43 
43 
43 

43 

44 

0.70461 To 
0.7089 .I 
0.7133 44 
0.7177 44 
0.7221 

- 44 

0.7265 


0.6536 

0.6577 

0.6619 

0.6661 

0.6703 


0.6745 


0.6787 

0.6830 

0.6873 

0.6916 

0.6959 

0.7002 


101 

100 

100 

98 

98 

97 

96 

95 

94 

93 

93 

92 

91 

90 

90 

89 

88 

87 

87 

85 

86 

84 


Cot. d. TanJ d. Sin. d. ' 0 


0 63 

50 
40 
30 
20 
10 

0 62 

50 
40 
30 
20 
10 


61 


0 

50 
40 
30 
20 

10 

0 60 

50 
40 
30 
20 
10 


0 

50 

40 

30 

20 

10 

0 

50 

40 

30 

20 

10 

0 

50 

40 

30 

20 

10 

0 1 

50 

40 

30 

20 

10 

0 

50 

40 

30 

20 

10 


59 


38 

3.8 

7.6 

11.4 

15.2 
19.0 
22.8 
26.6 

30.4 

34.2 


37 

3.7 

7.4 

11.1 

14.8 

18.5 
22.2 

25.9 

29.6 
33.3 




26 

25 

24 


1 

2.6 

2.5 

2.4 

58 

2 

5.2 

5.0 

4.8 

3 

7.8 

7.5 

7.2 


4 

10.4 

10.0 

9.6 


4 

13.0 

12.5 

12.0 


6 

15.6 

15.0 

14.4 


7 

18.2 

17.5 

16.8 


8 

20.8 

20.0 

19.2 

57 

9 

23.4 

22.5 

21.6 


23 

17 

16 


1 

2.3 

1.7 

1.6 


2 

4.6 

3.4 

3.2 


3 

6.9 

5.1 

4.8 


4 

9.2 

6.8 

6.4 

56 

5 

11.5 

8.5 

8.0 

6 

13.8 

10.2 

9.6 


7 

16.1 

11.9 

11.2 


8 

18.4 

13.6 

12.8 


9 

20.7 

15.3 

14.4 



15 

14 

13 


1 

1.5 

1.4 

1.3 

55 

2 

3.0 

2.8 

2.6 

3 

4.5 

4.2 

3.9 


4 

6.0 

5.6 

5.2 


5 

7.5 

7.0 

6.5 


6 

9.0 

8.4 

7.8 


7 

10.5 

9.8 

9.1 

54 

8 

12.0 

11.2 

10.4 

9 

13.5 

12.6! 

11.7 

O 


P. 

P. 



74 
































































































































o 

36 

37 

38 

39 

40 

41 

42 

43 

I 

44 

i 

< 

I 

45 


Tan. d. Cot. d. Cos. d. 

1.3764 


0.7265 

0.7310 

0.7353 

0.7400 

0.7445 

0.7490 


0.7536 


0.7581 

0.7627 

0.7673 

0.7720 

0.7766 

0.7813 

0.7860 

0.7907 

0.7954 

0.8002 

0.80jO 


0.8098 


0.8146 

0.819j 

0.8243 

0.8292 

0.8342 


45 
45 
45 
45 

45 

46 

45 

46 

46 

47 

46 

47 

47 
47 

47 

48 
48 

48 

48 

49 

48 

49 

50 

49 

50 
50 
50 

50 

51 

51 

0.8693 

- o 1 

0.8744 52 
0.8796 
0.8847 52 
0.8899 53 
0.8952 ^ 

53 
53 

53 

54 
54 

54 

55 
55 
55 

55 

56 

56 

56 

57 
57 

57 

58 

58 


0.8391 

0.8441 

0.8491 

0.8541 

0.8591 

0.8642 


0.9004 


0.9057 

0.9110 

0.9163 

0.9217 

0.9271 


0.9325 


0.9380 

0.9435 

0.9490 

0.9545 

0.9601 


0.9657 

0.9713 

0.9770 

0.9827 

0.9884 

0.9942 


1.0000 


1.3680 

1.3597 

1.3514 

1.3432 

1.3351 

1.3270 

1.3190 

1.3111 

1.3032 

1.2954 

1.2876 

1.2799 

1.2723 

1.2647 

1.2572 

1.2497 

1.2423 

1.2349 

1.2276 

1.2203 

1.2131 

1.2059 

1.1988 

1.1918 

1.1847 

1.1778 

1.1708 

1.1640 

1.1571 

1.1504 

1.1436 

1.1369 

1.1303 

1.1237 

1.1171 

1.1106 

1.1041 

1.0977 

1.0913 

1.08'0 

1.0786 

1.0724 

1.0661 

1.0599 

1.0538 

1.0477 

1.0416 

1.0355 

1.0295 

1.0235 

1.0176 

1.0117 

1.0058 


1.0000 


Cot. d. Tan. d. Sin. d. 


64 

63 


62 

63 

62 

61 

61 

61 

61 

60 

60 

59 

59 

59 

58 


0.8090 

3 0.8073 
j 0.8056 
, 0.8039 

J 0.8021 

0.8004 

0.T986 

, 0.7969 
0.7951 

3 0.7934 

3 0.7916 
0.7898 

0.7880 

5 0.7862 
0.7844 
0.7826 
0.7808 
0.7790 

0.7771 

1 0.7753 
0.7735 
0.7716 
0.7698 
0.7679 

0.7660 

0.7642 

0.7623 

0.7604 

0.7585 

0.7566 

0.7547 

0.7528 

0.7509 

0.7490 

0.7470 

0.7451 

0.7431 

0.7412 

0.7392 

0.7373 

0.7353 

0.7333 

0.7314 

0.7294 

0.7274 

0.7254 

0.7234 

0.7214 

0.7193 

0.7173 

0.7153 

0.7133 

0.7112 

0.7092 


1 

1 

17 

18 

17 

18 

17 

18 

17 

18 
18 

18 

18 

18 

18 

18 

18 

19 

18 

IS 

19 

18 

19 

19 

18 

19 

19 

19 

19 

19 

19 

19 

19 

20 

19 

20 

19 

20 

19 

20 
20 

19 

20 


0.7071 


0 54 
50 
40 
30 
20 
10 


53 


52 


51 


0 
50 
40 
30 
20 
10 

0 
50 
40 
30 
20 
10 

0 

50 
40 
30 
20 
10 

0 50 

50 
40 
30 
20 
10 

0 49 

50 
40 
30 
20 
10 


0 46 

50 

40 

30 

20 

10 

0 45 




P. I 




58 

57 

56 

55 

1 

5.8 

5.7 

5.6 

5.5 

2 

11.6 

11.4 

11.2 

11.0 

3 

17.4 

17.1 

16.8 

16.5 

4 

23.2 

22.8 

22.4 

22.0 

5 

29.0 

28.5 

28.0 

27.a 

6 

34.8 

34.2 

33.6 

33.0 

7 

10.6 

39.9 

39.2 

38.5 

8 

46.4 

45.6 

44.8 

44.0 

9 

52.2 

51.3 

50.4 

49.5 


54 

53 

52 

51 

1 

5.4 

5.3 

5.2 

5.1 

2 

10.8 

10.6 

10.4 

10.2 

3 

16.2 

15.9 

15,6 

15.3 

4 

21.6 

21.2 

20.H 

20 4 

5 

27.0 

26.5 

26.0 

25.5 

6 

"2.4 

31.8 

31.2 

30 6 

7 

37.8 

37.1 

36.4 

35.7 

8 

43.2 

42.4 

41.6 

40.H 

9 

48.6)47.7 

46.8 

45.9 


50 

5.0 

10.0 

15.0 

20.0 

25.0 

30.0 


49 48 

4.9 4.8 
9.8j 9.6 
14.7 i 14.4 
19.6 19.2 
24.5 24.0 
29.4;28.8 
35.0 34.3 33.6 
40.0 39.2 38.4 
45.0;44.1 43.2 

47 46 45 

4.7: 4.6 4.5 
9.41 9.2 j 9.0 
14.1 13.8 13.5 
18.8 18.4 j 18.0 
23.523.0 22.5 
28.2 [ 27.6 27.0 
32.9132.2 31.5 
37.6:36.8 36.0 
42.3 41.4 40.5 


v HO 

50 


24 

23 

40 

1 

2.4 

2.3 

30 

2 

4.8 

4.6 

20 

3 

7.2 

6.9 

10 

4 

9.6 

9.2 


5 

12.0 

11.5 

047 

6 

14.4 

13.8 

50 

7 

16.8 

16.1 

40 

8 

19.2 

18.4 

30 

9 

21.6 

20.71 

20 


20 

19 

10 

1 

2.0 

1.9 


22 

2.2 

4.4 
6.6 
8.8 
11.0,10.5 
13.2 12.6 
15.4 14.7 
17.6 16.8 
! 19.8 j 18.9 


21 

2.1 

4.2 

6.3 

8.4 


4.0 

6.0 

8.0 

10.0 


3.8 

5.7 

7.6 

9.5 


12.0111.4 
14.0 J3.8 
16.0115.2 

J 18.0117.1 


18 

1.8 
3.6 
5.4 
7.2 
9.0 
10.810.2 
12.6 11.9 
14.4 13.6 
16.2 15.3 


17 

1.7 

3.4 
5.1 

6.8 

8.5 


P. P. 





































































































76 


SURVEYING 


SURVEYING 

INSTRUMENTS USED 

THE COMPASS 

Surveying is an extension of mensuration, and, as ordinarily- 
practiced, may be divided into surface work, or ordinary sur¬ 
veying, and underground work, or mine surveying. With slight 
modifications, the instruments employed in both are the same. 

The compass used may be either a pocket compass, or a 
surveyor’s compass; it may be held in the hand, or on a tripod. 
The Jacob’s staff is convenient for use on the surface, but is 
frequently useless in the mine. The compass is not accurate 
enough for the construction of a general map of the mine. 
It may be used to secure an approximate idea of the shape 
of the workings, so as to plan an approximate course on which 
to drive an opening designed to connect two or more given 
points. If the opening is one that will be expensive to drive, 
and should be straight, the compass survey should never be 
relied on. 

Using the Compass. —In using the compass, the surveyor 
should keep the south end toward his person, and read the 
bearings from the north end of the needle, care being taken 
always to keep the compass level. In the surveyor’s compass, 
the position of the letters E and W are reversed from their 
natural position, in order that the direction of the sight may 
be correctly read. As the circle is graduated to 5 0 , a little prac¬ 
tice will enable the surveyor to read the bearings to quarters, 
estimating with his eye the space bisected by the point of the 
needle. 

The compass is usually divided into quadrants, and 0 is 
placed at the north and south ends; 90° is placed at the E and 
W marks, and the graduations run right and left from the 0 
to 90°. In reading the bearing, if the sights are pointed in a 
N W direction, the north end of the needle, which always points 
approximately north, is to the right of the front sight or front 



SURVEYING 


77 


end of the telescope, and, as the number of degrees is read 
from it, the letters marking the cardinal points of the compass 
read correctly. If the E, or east, mark were on the right side 
of the circle, a N W course would read N E. This same 
fact applies to all four quadrants. 

THE TRANSIT 

The transit is the only instrument that should be used for 
measuring angles in any survey where great accuracy is desired. 
Its advantages over a vernier compass are mainly due to the 
use of a telescope. With it angles can be measured either 
vertically or horizontally, and, as the vernier is used through¬ 
out, extreme accuracy is secured. In mine work all screws 
and movable parts should be covered, so as to keep out acid, 
water, and dust; if this is not done, the instrument will soon 
be destroyed. The vertical circle on the transit may be a 
full circle or a segment; the former is preferred, as it is always 
ready without intermediate clamp screws. 



Transit Verniers. —The verniers on a transit differ from those 
on a compass in detail only; the principle is the same. The 
transit vernier is so divided that 30 spaces on it equal in length 
29 on the limb of the instrument. It is read practically the 
same as a compass vernier, except that on the transit the vernier 
is made with all of the 30 divisions on one side of the 0 mark. 

Each division of the vernier is, therefore, -yfr, or, in other 
words, V shorter than the 5 0 graduations on the limb. In 
the figure the reading is 20° 10'. If the 0 on the vernier should 
be beyond 20 j° on the limb of the transit, and the line 10 should 


























78 


SURVEYING 


coincide with a line on the limb, the reading would be 20° 40'. 
In case the 12th line from 0 should coincide with a line on the 
limb, the reading would be 20° 42', etc. 

In some transits, the graduated limb has two sets of con¬ 
centric graduations, the 0 in both being the same. While 
the outside set is marked from 0° each way to 90°, and thence 
to 0° on the opposite side of the circle, the other set is marked 
from 0° to 360° to the right, as a clock face. The inside set 
has the N, S, E, and W points marked, the 0° of the inside 
set being taken as north. 

Transit Telescopes. —The interior of the telescope is fitted 
with a diaphragm, or cross-wire ring, to which cross-wires 
are attached. These cross-wires are either platinum or strands 
of spider web. For inside work, platinum should be used, as 
spider web is translucent and cannot readily be seen. These 
wires are set at right angles to each other and are so arranged 
that one can be adjusted so as to be vertical and the other 
horizontal. This diaphragm is suspended in the telescope 
by four capstan-headed screws, and can be moved in either 
direction by working the screws with an ordinary adjusting pin. 
The intersection of the wires forms a very minute point, that, 
when adjusted, determines the optical axis of the telescope, 
and enables the surveyor to fix it upon an object with the 
greatest precision. 

The imaginary line passing through the optical axis of the 
telescope is termed the line of collimation, and the operation 
of bringing the intersection of the wires into the optical axis 
is called the adjustment of the line of collimation. 

The transit should not be subjected to sudden changes in 
temperature that may break the cross-hairs. In case of a 
break, the cross-hair diaphragm must be removed and the 
broken wire replaced. 

CHAIN, TAPE, PINS, AND PLUMB-BOB 

The chain, which is generally 50 or 100 ft. long, should be 
made of annealed steel wire, each link being exactly 1 ft. in 
length. 

The steel tape is simply a ribbon of steel, on which are marked, 
by etching, or other means, the different graduations; these 


SURVEYING 


79 


may be inches or tenths of a foot, or every foot. It is wound 
on a reel, and may be any desired length up to 500 ft. When 
distances do not come at even feet, the fractional part of the 
foot should always be noted in tenths. Thus, 53 ft. 6 in. should 
always be noted as 53.5 ft. 

For the most exact work steel tapes are now almost exclusively 
used by the leading mining engineers, on account of their greater 
accuracy as compared with chains. 

Pins should be from 15 to 18 in. long, made of tempered- 
steel wire, and should be pointed at one end and turned with 
a ring for a handle. 

The plumb-bob takes the place of the transit rod in under¬ 
ground work, as the stations are usually in the roof, and strings 
are hung from them to furnish foresights and backsights. 
Plumb-bobs vary in weight and shape. The cord is best 
illuminated by placing white paper or cardboard behind it 
and holding the lamp in front and to one side. The string 
shows as a dark line against a white ground, and there is less 
difficulty in finding it than when the light is placed exactly 
behind it. 

The clinometer, or slope level, is a valuable instrument for 
side-note work, but it is not accurate enough for a survey; 
its place has been taken by the vertical circle on the transit. 
Clinometers are of two styles, one showing the inclination by 
means of a bubble and the other, by means of a pendulum. 
The latter is the old-fashioned and more accurate German 
Gradbogen. 


TRANSIT SURVEYING 


READING ANGLES 

The angle read may be included or de¬ 
flected. If the transit is set up at O with 
backsight at B and foresight at C, there -B 

are two angles made by the line C O with 
the line BOA, namely the included angle 
B O C, and the deflected angle CO A. 

To read the included angle set the zeros of the vernier and 
graduated limb together accurately, and clamp the plates. 






89 


SURVEYING 


Turn the telescope on the backsight, with the level bubble 
down, and, when set, fasten the lower clamp so as to fix both 
clamped plates to the tripod head. Loosen the upper clamp, 
turn the telescope to C, and set accurately. The vernier will 
read, for example, 45° left angle. 

To read the deflected angle arrange the verniers as before, 
being careful to turn the telescope over on its axis until the 
bubble tube is up, and then take the backsight and fix lower 
clamp. Turn the telescope back (this is called -plunging the 
telescope) and then sight to foresight and fix as before; the 
vernier will then read a right angle of 135°. The sum of 
included and deflected angles must always be 180°. 

Note. —In making a survey by included angles it is neces¬ 
sary to add or subtract 180° at each reading to have the vernier 
and compass agree; by deflected angles they will agree without 
the above addition or subtraction, therefore the latter method 
is generally used. 

If the dip of a sight is to be taken the tape must be held 
at the transit head and stretched in the line of sight. If the 
pitch of the ground is to be taken the point of foresight must 
be at the same height as the axis of the transit and the sight 
will then be parallel to the surface. The angle of dip is read 
“plus” or “minus” as it is above or below the horizontal 
plane. If we have the dip of a sight and the distance between 
the transit head and the point of sight we can get the vertical 
and horizontal components of that distance from the table of 
sines and cosines. 

MAKING SURVEY WITH TRANSIT 

Surveying by Means of Individual Angles.—To survey by 
means of individual angles, set the vernier at 0 of limb, plunge 
the telescope, and, when set on the backsight, loosen the needle 
and read the bearing of the line from backsight to set-up. 
Plunge the telescope back and set on the foresight and read 
both the needle and the vernier. The difference in the needle 
readings should agree with the vernier reading within 15', as 
local attraction will affect the needle equally on both sights. 

Note. —As the moving of any mass of iron or steel during 
the readings of the needle will affect the same and destroy the 
value of the needle as a check, the tape and other iron mate¬ 
rials should not be moved during the taking of angles. 

Surveying by Means of Continuous Vernier.—To survey by 
means of continuous angles, set the vernier at 0, unclamp the 


SURVEYING 


81 


compass needle, and, when stationary, turn the north point of 
the compass limb so as to coincide with the north point of the 
needle. Fix the lower clamp, plunge the telescope, and take 
a backsight by loosening the upper clamp. The vernier and 
the needle should agree in giving the magnetic bearing of the 
line from backsight to set-up. Record this in the notebook; 
plunge the telescope, and take a foresight; the needle and 
the vernier should agree as before. After making the record, 
set up over the foresight and take a sight to the station just 
left with the telescope plunged, having first seen that the 
vernier reads exactly as it did on the last foresight, as a slip in 
carrying the transit from one station to another, which is not 
detected at the time, can never be checked afterwards when 
the final work is found to be in error. The foresight is taken 
as before; on every sight the needle and the vernier should 
agree if there is no local attraction of the needle. 

If all the corners of a field that is to be surveyed can be'seen, 
from a central point, the survey can be made by setting up 
the transit at that point, and, with one corner as a backsight, 
taking all the other comers as foresights with but one set-up, 
and measuring from this point to all of the corners; or the transit 
can be set up at any corner and a line run around the field. 
This latter method is called meandering. Both methods will 
give the same result when plotted. 

PLOTTING 

A plot is a map drawn to a given scale, and showing all of 
the natural features. Plotting is the making of such a map 
from notes of a survey, and may or may not require the per¬ 
manent placing on it of the stations, by which the survey 
is made. In underground work, the exact location and the 
retention of those stations is of the first importance, and is 
secondary only to the exact plotting of the side notes. The 
scale of the plot is generally as large as will show the points of 
interest in the property; but in Pennsylvania, the maps for 
coal mines must be drawn to a scale of 100 ft. = 1 in. There 
are two methods of plotting: by protractor, and by coordi¬ 
nates. When the scale is sufficiently large, it makes little 
difference which method is used, if the work be carefully done 



82 


SURVEYING 


with exact instruments; but with small scales, 100 ft. = 1 in., 
or smaller, the method by coordinates should be used. When 
the scale is from 1 to 25 ft. to 1 in., the errors are small enough 
to make little chances of variation in a close of ten or twelve 
stations; when the survey is of short sights from a main line 
to points where no further work is to be done, the protractor 
will afford a quick method of plotting. 

To Calculate Vertical Distances. —When making the survey, 
read the vertical angles to all stations. If the angle is one of 
depression, place a minus sign ( —) before it; if it is an angle 
of elevation, place a plus sign (-J-) before it. These will show 
whether the vertical distance is to be added to, or subtracted 
from the height of the preceding station. 

Having the horizontal distance and the vertical angle: 

Distance X tangent of vertical angle = vertical distance. 

Having the pitch distance and vertical angle: 

Distance X sine of vertical angle = vertical distance. 

To Calculate Horizontal Distance, or Latitude.—Pitch dis¬ 
tance X cosine of vertical angle = horizontal distance. 

Vertical height or departure -f-tangent of vertical angle 
= horizontal distance. 

To Calculate Pitch Distance.— Horizontal distance 4- cosine 
of bearing, or multiplied by secant of bearing = pitch distance. 

Vertical distance-f-sine of vertical angle, or multiplied by 
cosecant of bearing = pitch distance. 

To Calculate Vertical Angle. —Horizontal distance -5- the 
pitch distance = cosine of vertical angle. 

Vertical distance -5- pitch distance = sine of vertical angle. 

Vertical distance 4- horizontal distance = tangent of vertical 
angle. 

Note.— Whenever sines, cosines, tangents, etc., are here 
named, they mean the natural sines, etc. of the angle. 

Plotting by Coordinates. —In the establishment of a meridian 
and a fixed point, the latter should be a stone post, or iron plug 
sunk in solid rock; this point is called the origin of coordinates. 
Have the principal meridian passing through this point in an 
exact north-and-south direction, and a secondary meridian or 
base line passing through this point at right angles to the first, 
or in an exact east-and-west line. Any point on the map will 


SURVEYING 


83 


then be a certain distance north or south, and east or west of 
the origin. The lines drawn from this point at right angles 
to the two base lines just given are called the coordinates of 
that point, and the point can be plotted when they are given. 
For example, the coordinates of a station are N 345.67, and 
E 890.12. Measure 890.12 ft. east of the origin on the second¬ 
ary meridian and, from this point, 345.67 ft. north to the point 
desired. Or measure on the primary meridian to the north 
and then turn off a right angle to the east and reach the same 
point. In any event the position of each station may be plotted 
independently of all the others, and any error in locating one 
is not carried to the next. When two stations are plotted, the 
distance between them on the map should be exactly what is 
found for their horizontal distance on the ground. This check 
shows whether the plotting is correct. This is also called 
traversing a survey if the meridian is north and south, and in 
books on surveying there are printed traverse tables, which are 
accurate within certain limits, but not so accurate as the tables of 
coordinates published separately, as the latter are carried to a 
greater number of decimals. 

With a north-and-south meridian, the point from which the 
measuring of the angles is begun, the zero point, is the north 
point, and the angles are read for continuous vernier in 
the direction of the hands of a watch. The sines of angles 
are eastings and westings, and the cosines are northings and 
southings. 


TRAVERSING A SURVEY 

To traverse a survey, means to determine by calculation 
how far north or south and east or west any station may be 
from another, the location of which is fixed. To do this, 
all distances must be measured horizontally, or calculated to 
horizontal distances. The horizontal angles, or courses, must 
be read as quadrant courses, or reduced from azimuth to quad¬ 
rant courses. An azimuth course is one that is read on a transit 
that is graduated from 0° to 360°. A quadrant course is one 
read in the quadrant of the circle, as S 67° W, N 43° E, etc. 

Latitude means distance north or south, and is determined 
by the first initial of the recorded course; thus, if a course is 


84 


SURVEYING 


S 67° W, the latitude is south; if N 43° E, the latitude is north. 
The latitude = distance X cosine of bearing. 

Departure means distance east or west, and is determined 
by the last initial of the recorded course; thus, if a course is 
S 67° W, the departure is west; if N 43° E, the departure is 
east. The departure = distanceX sine of bearing. 

If the survey is a continuous one around a tract, and ending 
at the place of beginning, the sum of the northings should equal 
the sum of the southings, and the sum of the eastings should 
equal the sum of the westings. The most accurate way to 
construct a map is to traverse the survey and place all stations 
on it by the traversed distances, or to at least put a number of 
the principal stations on the map by the traversed distances, 
and use the protractor to plot only the intermediate stations. 

DETERMINING AREA OF TRACT OF LAND 

If the tract of land is a regular polygon, find the area by 
the rule given under the head of Mensuration for polygons of 
the same number of sides. If it is an irregular polygon, divide 
it into triangles and calculate the area of each triangle; the 
sum of these areas will be the area of the tract. If the tract 
is an irregular polygon in shape, the map should be made on as 
large a scale as possible, and the distances should be measured 
with the greatest care, owing to liability to error through very 
slight inaccuracies of measurement. 

DETERMINING CONTENTS OF COAL SEAM 

If the seam lies flat, multiply the area of the tract, in square 
feet, by the thickness of the seam, in feet; the product will be 
the cubic contents of the seam, in feet. If the seam is an 
inclined one, find its area by measuring the width of the tract 
on its line of pitch, and find the distance on the pitch of the 
seam by dividing the horizontal distance measured by the 
cosine of the angle of inclination; this will give the pitch dis¬ 
tance. Multiply the pitch distance by the length of the tract, 
to find the area of the seam; this multiplied by its thickness 
will give the contents. 

cubic contents, in feet Xsp. gr.X 62.5 

Tons of coal =--—- 


2,240 



SURVEYING 


85 


UNDERGROUND SURVEYING 

ESTABLISHMENT OF STATIONS 

There are a number of variations in the foregoing practice 
that are caused by the entirely different set of conditions in 
underground work. As the establishment of stations is the 
most important duty of an engineer in surface work, so it 
takes the first place in work underground, as the accuracy 
of the work depends on the location of the stations, while its 
rapidity depends on using the least number consistent with 
completeness. Also, the fewer the number of stations, the 
less are the chances of error. In underground work, stations 
should be located under the conditions of permanence, freedom 
from destroying agencies, and ease of access. Temporary 
stations for a single sight need not fill all these requirements. 
They are generally established in the roof of the mine, less 
frequently in the floor. In the former case, a center must be 
established before each set-up of the transit. Places are chosen 
that will be least affected by subsequent work, and the stations 
are put in collars, lids, or wedges of props, in the props themselves 
when they have sufficient incline to allow the transit to be set 
under them, or in the roof itself. Wherever set, they should 
not project far from the surface, and thus be liable to be brushed 
away in a low gangway by cars with topping higher than usual, 
or knocked away by flying fragments from a shot, if near the 
working faces. Top stations have a mark about them to call 
attention to their location; it is generally a circle. When there 
are other corps at work in the same mine, the stations of the 
two surveys should be given distinguishing marks to avoid 
confusion. 

Kinds of Stations.—The simplest top station is called by 
some a jigger station. It is a shallow conical hole, made with 
the point of the foresight man’s hatchet which is dug into the 
top rock and rotated. The sights are given and the centers 
set by putting the plummet cord in this groove, and placing 
the end in the jigger hole in the roof. Common shingle nails 
are sometimes driven into collars, or cracks in the roof, and 
the end of the plummet line is noosed and put over the head. 


86 


SURVEYING 


This causes an eccentric hanging of the plummet that may result 
in an error in backsight and foresight of the width of the nail 
head, which will be quite appreciable in a short sight. 

A wooden plug is driven into a hole drilled in the roof, and 
into this is driven the spad. The swelling wood clamps the 
same and prevents it from coming out as readily as it was 
put in. 

A hole is bored in the roof with a A-in. twist drill and a 
piece of cord or a copper wire, placed across this and driven 

into the hole by a hardwood shoe peg. The plummet is tied 

to the lower end. A cord will soon rot, and, if in the gang¬ 
way, will be pulled out by the drivers for whip lashes; while 
the wire is more permanent, it may be pulled out by catching 
in the topping of a car in a low place. 

In the best form of station, however, the use of spads is 
dispensed with and all the stations are put in rock roof where 
possible, and consist of a vertical hole 1 in. deep made with 

a A-in. twist drill. When a sight is to be taken, the foresight 

man puts into this a steel clip with serrated edges; this clip is 
made by bending upon itself a thin piece of steel A -in. wide. 
When the ends are pressed together it will go into the hole, 
and the spring of the sides and the serrated edges hold it in 
place so that it is hard to pull out. The cord passes through a 
hole in the center of the bend and is, therefore, in the center 
of the hole, no matter how the clip is inserted. The clip is 
removed by pressing together the ends. This is the easiest 
and quickest way of working, as there is no eyehole to be freed 
from dirt and no knot to be tied and untied. The hanging 
of the plummet takes a fraction of a second, and the station will 
remain as long as the roof keeps up. The disadvantages are 
that holes may be bored inclined to the vertical by a careless 
man, and many roofs are unfit for piercing with a twist drill. 

Marking Stations.—There should be some regular way of 
witnessing all stations. In general, a vertical line on the rib 
calls attention to a station in the floor near the side marked. 
A roof station has a mark around it, as has been described, 
and is some geometric figure. Each station must be lettered 
or numbered so that it can be readily recognized when the 
subsequent surveys are made. 


SURVEYING 


87 


White lead, or Dutch white, thinned with linseed oil, is 
ordinarily used for marking stations. The top should be wiped 
clean and dry with a piece of cotton waste before the paint is 
applied, or the white will be so discolored as to be scarcely 
visible, or the paint will flake off and the numbers will be lost. 

Centers.—When the station is in the roof, there must be 
something for the transit to set over, as it is easier to do so 
than to set under a station, and much more accurate as instru¬ 
ments are now made. The set-up is made over a center. 
To avoid being displaced, centers are made as small and heavy 
as possible; they are usually made of lead in the following 
manner: A hole lj in. in diameter and 5 in. deep is bored in 
a thick plank, and a brad is set in its center with the head 
down. The hole is filled with melted lead and the brad is 
slightly raised to surround the head with lead, and held with 
pincers in a vertical position until the lead has set. The brad 
is cut off i in. above the lead and pointed. This center com¬ 
bines weight and small size, 

KEEPING NOTES 

Taking Notes.—Complete notes should be taken and recorded 
neatly and systematically, so that a stranger can easily follow 
them. Every physical characteristic and all surface improve¬ 
ments should be noted and located. Every ledge of rock 
should be noted, its character, dip, and course of strike should 
be taken. In a large company there should be a separate 
book for transit notes and for side notes, and where many 
collieries are operated, a separate set of books should be used 
for each colliery. But however the notes are kept, the follow¬ 
ing facts should be recorded: The numbers of the stations; the 
needle readings to check the vernier; the vernier reading; the 
dip of the sight; the distance measured, either flat or on the 
dip; the height of the axis of the transit from the ground; 
the height of the point sighted at from the ground; and all 
other necessary remarks to make the work plain. It is custom¬ 
ary to have series of vertical columns headed (to suit the above) 
Sta., Needle F. S., Needle B. S., Vernier, Pitch, Dist., H. I. 
(height of instrument), H. R. (height of rod, or point to which 
sight was taken), and Remarks. 


88 


SURVEYING 


At the top of the page, in starting a survey, there should be 
entered the name of the mine and of the bed where the 
work is to be done; the names of the regular corps employed 
for the work, and those that were taken from the mine to 
point out work or assist; the instruments used; the date of 
the work, and, in case it is the continuation of a previous 
survey, the pages where such work was noted must be set 
down. Such books are complete records, and can be used 
as time books in paying the men, or as proofs of the kind 
of work done in case a lawsuit requires such testimony, by 
showing the number of men, the instruments used, and the 
time employed. 

Transit and Side Notes.—There are about as many methods 
of keeping the transit and side notes as there are engineers. 
These methods arrange themselves into groups; those in most 
common use in the mines are: 

The side notes of each sight follow the transit notes of that 
sight, and on the same page. 

The side notes are entered in the same book on opposite 
pages. 

The transit notes of the whole survey come first, and are 
followed by the side notes in the same book. 

Each set of notes has a separate book. 

Suppose that the transit is set up at b, with backsight at 
a; foresight to c, deflected angle abc = 85° 27' left, and that 
the distance b c is 421.76 ft. measured on a pitch of +4° 35'. 
Then some such form of notes as the following can be used. 
Other forms are used, but all are made to suit the ideas of the 
person surveying. 


Sta. 

Needle 
B. S. 

Vernier 

Needle 
F. S. 

Pitch 

Dist. 

A 

B 

— a 
b 

S 25° 
30'W 

L 85° 
27' 

L 85° 
26' 

S 60° 

0' E 

+4° 35' 

421.76 























SURVEYING 


89 


In every case the notes should convey to the man that plots 
some idea of the form of the place surveyed. An accurate 
sketch cannot be made unless the whole locality can be seen 
at a glance; it is not necessary to go to the other extreme 
and write down the notes without a sketch. 

In the accompanying illustration, the red line in the 
center of the page of the notebook is taken as the line of 
survey, and the next parallel line on either side is taken 
as the boundaries of the solid on either side. The L only 
figures on each side of the red line 
are the distances from the line to 
the solid, while the pluses at which 
they were taken are noted at the 
side of the page, and the exact dis¬ 
tance between the two stations is 
enclosed in the parallelogram. This 
method at the pluses 155 and 157 
calls attention to a point where 
practice varies greatly; namely, the 
noting of the comer pillar and the 
locating of the comer. One method 
calls the corner that point where 
the pillar begins to diverge from 
the gangway line, as at a, where a 
chamber, cross-cut, or counter 
starts from the gangway; a second 
method designates the corner as 
the first or last solid part met with 
in the line of survey, as at b. The 
first is faulty, as there is no record 
of the gradual divergence of the pillar from the gangway line, 
and the words “corner of pillar” usually mean the end of the 
same. The pillar should be called solid until the line at right 
angles to the line of survey is tangent to the ends, no matter 
whether that end is 10 or 100 ft. distant. Any one can plot 
side notes if accurately taken, and two persons accurately 
plotting such notes will reach the same result. 














90 


SURVEYING 


MINE CORPS 

The method of dividing the work in an underground survey 
depends on the size of the corps, therefore the work of each 
man is considered in order to get the right number for the corps. 
The chief of the party should be where he can do the most 
good, and where he can plan the work for his subordinates. 
The principal point of the survey is the setting of the stations 
so as to do the work thoroughly with the fewest set-ups, and 
thus diminish the chances of error in instrument work. The 
chief should locate the stations and add all the necessary signs 
to show how the work is to be done. As the transilman 
should not have his attention distracted from his particular 
work by questions as to procedure, the chief should not run 
the transit. Upon this basis, the ideal mine corps consists of at 
least four, or better five, men from the office, and three from 
the mine. It is divided into two sections. The chief takes 
the men supplied by the mine—one or more of whom is 
acquainted with the work done since the last survey—and 
locates the stations; the transitman follows with the second 
section, to measure angles and distances. By this time the 
stations are set and the chief takes his men after the transit 
party and gets the side notes, with a check-measurement of the 
distances between stations. 

The foresight man should be intelligent and active, as the 
amount of work done in a day depends on his ability to keep 
ahead of the transitman. Some of the latter are fast enough 
to keep two foresight men on the jump. His duty is to set 
the center for the next set-up under the station, and also place 
the tripod if three are used in the work, to give the sight, 
and, in some corps, to carry the front end of the tape and 
assist in taking the distance. 

The backsight man has little to do inside; for this reason, he 
cleans and oils the tape, gets out new plummet strings, and 
sees that the tools are ready for the next work, as soon as 
the corps gets to the office. 

LAYING OUT CURVES IN A MINE 

Curves in a mine are usually so sharp that they are desig¬ 
nated as curves of so many feet radius, instead of as curves 


SURVEYING 


91 


of so many degrees. For example, suppose that it is required 
to connect the two headings A and B, in the accompanying 
illustration, which are perpendicular to each other, with a 
curve of 60 ft. radius. Prepare the device shown on the 
right-hand side, by taking three small wires or inelastic 
strings f g, g h, and g k, each 10 ft. long, and connecting 
one end of each to a small ring, and the other end of two 
to the ends of a piece of wood If ft. long. Form a neat 
loop at the end / of the 
third gf. To use this de¬ 
vice, lay off on the center 
line of the heading B, c d 
and d e equal to 60 ft. 
and 10 ft., respectively. 

Place the loop / of the de¬ 
vice over a small wire peg 
driven into the floor at e, 
and the ring g over a simi¬ 
lar peg at d. Take hold 
of the stick h k, pull the 
strings g h and g k taut, and place the center mark on the piece of 
wood hk on the center line of the heading B. Drive a small peg 
in at m, located by the point k, which is on the curve. Move 
the device forwards, place the loop/over the peg at d, the ring 
g over the peg at m, and take hold of the stick h k and pull 
until the strings g h and g k are taut, and the strings / g and g h 
are in a straight line. The point k will fall on the curve at n , 
which mark by driving in a peg. To locate other points, 
proceed exactly as in the last step. The distance c d in any 
case is found by the formula 

c d = R tan § / 

in which R = radius of curve; 

I = intersection angle of center lines of headings 





















92 


ELEMENTS OF MECHANICS 


ELEMENTS OF MECHANICS 


DEFINITIONS AND LAWS 

All machinery, however complicated, is merely a combina¬ 
tion of the six elementary forms: the lever, the wheel and axle, 
the pulley, the inclined plane, the wedge, and the screw; and 
these can be still further reduced to the lever and the inclined 
plane. They are termed mechanical powers, but they do not 
produce force; they are only methods of applying and direct¬ 
ing it. The law of all mechanics is: 

Law.— The power multiplied by the distance through which it 
moves is equal to the weight multiplied by the distance through 
which it moves. 

Thus, 20 lb. of power moving through 5 ft. = 100 lb. of weight 
moving through 1 ft. In the following discussion friction is 
not considered, the idea being to give an elementary knowledge 
of the principles of the elements of mechanics. 

Levers.—There are three classes of levers. They are: 

(1) power at one end, weight at the other, and fulcrum between; 

(2) power at one end, fulcrum at the other, and weight between; 

(3) weight at one end, fulcrum at the other, and power between. 
A lever is in equilibrium when the arms balance each other. 
The distances through which the power and the weight move 
depend on the comparative length of the arms. Let L repre¬ 
sent power’s distance from the fulcrum (C), l the weight’s 
distance, and a the distance between power and weight; then, 
if L is twice l, the power will move twice as far as the weight, 
Substituting these terms in the law of mechanics, 

l:L PL = Wl 

Wl PL 

— W - 

L l 

Pa Wa 

- L = - 

W+P W+P 



P:W = 
P = 

1 = 







ELEMENTS OF MECHANICS 


93 



P:W=l:L 

Wl 
P = — 
L 


Wa 

L = - 

W-P 


PL — Wl 

PL 
W= — 
l 


Pa 

1 = - 

W-P 



In first- and second-class levers, as ordinarily used, power 
is gained but time is lost; and in the third class, time is gained 
but power is lost. 

Example. —Having to lift a weight of 2,000 lb. with a lever, 
the short end of which is 2 ft. from the fulcrum and the long 
end 10 ft., how much power will be required? 

Solution. —Substituting in the formula L :l = P :W and 
2,000X2 

solving gives -——-= 400 lb. 

Wheel and Axle.—The wheel and axle, Fig. 1, of which the 
ordinary windlass is a common form, is a modification of the 
lever. The power is applied to the handle, the bucket is the 
weight, and the axis of the windlass is the fulcrum. The 
long arm of the lever is the handle, and the short arm is the 
radius of the axle. Thus, F is the fulcrum, 

F c the long arm, and F b the short arm. 

The wheel and axle has the advantage 
that it is a kind of perpetual lever; it is 
not necessary to prop up the weight and 
readjust the lever, but both arms work 
continuously. 

By turning the handle or wheel around 
once, the rope will be wound once around 
the axle, and the weight will be lifted 
that distance. Applying the law of mechanics, powerX 
circumference of wheel = weightXcircumference of axle; or, 






















94 


ELEMENTS OF MECHANICS 


as the circumferences of circles are proportional to their radii, 

P:W=r:R PR= Wr 

- P 

^ P = 



Wr 


R 

RP 


Wr 
R =— 
P 


RP 


Fig. 2 

one that receives motion from the driver. 


W = - r = 

r W 

A train, Fig. 2, con¬ 
sists of a series of wheels 
and axles that act on 
one another on the 
principle of a compound 
lever. The driver is 
the wheel to which 
power is applied. The 
driven, or follower, is the 
The pinion is the 


small gear-wheel on the axle. 

If the diameter of the wheel A is 16 in., and of the pinion B 
4 in., a pull of 1 lb. applied at P will exert a force of 4 lb. on 
the wheel C. If the diameter of C is 6 in., and of D 3 in., a 
force of 4 lb. on C will exert a force of 8 lb. on E. If £ is 16 in. 
in diameter, and F 4 in., a force of 8 lb. on E will raise a weight 
of 32 lb. on F. In order, however, to lift this amount according 
to the principle already named, the weight will only pass 
through 3 V of the distance of the power. Thus, power is 
gained and speed lost. To reverse this, apply power to the 
axle F, and, with a correspondingly heavy power, gain speed. 
Referring to Fig. 3, and applying the law of mechanics, the 
following formulas are obtained, 


P = 


Wrr'r" 

RR'R" 


W = - 


PRR'R" 


rr r 


n:n" = r'r" : RR' 


v:i/ = rr'r" : RR'R" 

In which «, n" = number of revolutions; 

v, v' = velocity of speed of 


r, r, r 
R, R', R" 


rotation; 

, etc. = radii of the pinions; 
etc. = radii of the wheels. 
















ELEMENTS OF MECHANICS 


95 


Inclined Planes.—In the inclined plane, Fig. 4, the power 
must descend a distance equal to A C to elevate the weight 
to the height B C; hence P X length of inclined plane = W 
X height of inclined plane, or P : W 
= height of inclined plane : length of 
inclined plane; or, 

Wh PI P 

P = - W=— =- 

l h sin a 



Wedge.—The wedge usually con¬ 
sists of two inclined planes placed 

back to back. In theory, the same formula applies to the 
wedge as to the inclined plane. 

P : W =thickness of wedge : length of wedge 
Screws.—The screw consists of an inclined plane wound 
around a cylinder. The inclined plane forms the thread, and 
the cylinder, the body. It works in a nut that is fitted with 
reverse threads to move on the thread of the screw. The nut 
may run on the screw, or the screw in the nut. The power 
may be applied to either, as desired, by means of a wrench or 
a lever. 

Pulleys.—The pulley is simply another form of the lever 
that turns about a fixed axis or fulcrum. With a single fixed 
pulley, shown in Fig. 5, there can 
be no gain of power or speed, as the 
force P must pull down as much 
as the weight W, and both move 
with the same velocity. It is sim¬ 
ply a lever of the first class with 
equal arms, and is used to change 
the direction of the force. v = ve¬ 
locity of W; v'= velocity of P; 

P= W; v = v f . 




Fig. 5 


Fig. 6 


In the movable pulley, shown in Fig. 6, one-half of the weight 
is sustained by the hook, and the other half by the power. As 
the power is only one-half the weight, it must move through 
twice the space; in other words, by taking twice the time, 
it is possible to lift twice as much. Here power is gained and 
time lost. P=\W\ v' = 2v. 












96 


ELEMENTS OF MECHANICS 


In the combination pulley, shown in Fig. 7, the weight W 
is sustained by three cords, each of which is stretched by a 
tension equal to the power P, hence, 1 lb. of power will balance 



8 4 21 2 




3 lb. of weight. In the combination shown in Fig. 8, 1 lb. will 
sustain a weight W of 4 lb. but it must descend 4 in. to raise 
the weight 1 in. Fig. 9 represents the ordinary tackle block 
used by mechanics. The power applied for balance can be 
calculated by the following general rule: 

Rule .—In any combination of pulleys where one con¬ 
tinuous rope is used, a load on the free end will balance 
a weight on the movable block as many times as great 
as the load on the free end as there are parts of the 
rope supporting the load, not counting the free end. 

In the combination shown in Fig. 10, each cord 
marked 1 has a tension equal to the power P; each 
cord marked 2, has a tension equal to 2 times P ; and 
so on with the other cords. As the sum of the tensions acting 
on the weight W is 16, IF=16P. If n = number of pulleys. 




v_y 

Fig. 11 


w 

P= —; W = 2 n P. 

2 n 

In the differential pulley, shown in Fig. 11, W= 


2 PR 
R-r 










































ELEMENTS OF MECHANICS 


97 


FRICTION AND LUBRICATION 

Friction.—Friction is the resistance to motion due to the 
contact of surfaces; it is of two kinds, sliding and rolling. 
If the surface of a body could be made perfectly smooth, 
there would be no friction; but, in spite of the most perfect 
polish, the microscope reveals minute projections and cavities. 
As no surface can be made perfectly smooth, some separation 
of the two bodies must, in all cases, take place in order to clear 
such projections as exist on the surfaces. Therefore, friction 
is always more or less affected by the amount of the perpen¬ 
dicular pressure that tends to keep them together. The 
ultimate friction is the greatest frictional resistance that one 
body sliding over another is capable of opposing to any sliding 
force when at rest. 

The coefficient of friction is the proportion that the ultimate 
friction in a given case bears to the perpendicular pressure. 
The coefficient of friction is usually expressed in decimals; 
but sometimes, as in the case of cars and engines, it is expressed 
in pounds (of friction) per ton. The coefficient of friction 
equals the ultimate friction divided by the perpendicular 
pressure, and the ultimate friction equals the perpendicular 
pressure multiplied by the coefficient of friction. Thus, if a 
block weighing 100 lb., stands on another block, and it takes 
a pressure of 35 lb. to slide it, the coefficient of friction = -Too, 
or .35. 

Lubrication.—To diminish the friction, oil or grease is placed 
on the surfaces of sliding bodies so as to fill the cavities and 
spaces between the projections; this oil or grease is called a 
lubricant. There is probably no factor that has a more direct 
bearing on the cost of production per ton of coal and ores than 
the lubrication of mine machinery, and yet it is doubtful if 
there is another item connected with the operation of a mine 
less understood by owners, managers, and engineers in charge. 


BEST LUBRICANTS FOR 

Low temperatures, as in rock 
drills driven by compressed 


air 


DIFFERENT PURPOSES 


Light mineral lubricating oils. 





98 


STRENGTH OF MATERIALS 


Very great 

speed. 

Heavy pressures, 

speed. 

Heavy pressures 
speed. 


with 


i 


and high 


pressures, slow j Graphite, soapstone, and 

.\ other solid lubricants. 

slow / The above, and lard, tallow, 
and other greases. 

Sperm oil, castor oil, and 
heavy mineral oils. 

Sperm, refined petroleum, 
olive, rape, cottonseed. 
Lard oil, tallow oil, heavy 
mineral oils, and the heav¬ 
ier vegetable oils. 

Heavy mineral oils, lard, tal¬ 
low. 

Clarified sperm, neat’s foot, 
porpoise, olive, and light 
mineral lubricating oils. 
For mixture with mineral oils, sperm is best; lard is much 
used; olive and cottonseed are good. 


Light pressures and high speed 


Ordinary machinery. 


Steam cylinders. 


Watches and 
mechanism. 


other delicate 


STRENGTH OF MATERIALS 


USEFUL FORMULAS 

The ultimate strengths of different materials vary greatly 
from the average values given in the following tables. In 
actual practice, the safest procedure is to make a test of the 
material for its ultimate strength and coefficient of elasticity, 
or else specify in the contract that it shall not fall below certain 
prescribed limits. In the following formulas, 

A =area of cross-section of material, in square inches; 

E = coefficient of elasticity, in pounds per square inch; 

G 2 = square of least radius of gyration; 

I = moment of inertia about an axis passing through center 
of gravity of cross-section; 

M = maximum bending moment, in inch-pounds; 

P = total stress, in pounds; 

R = moment of resistance; 












STRENGTH OF MATERIALS 


99 


5 = ultimate stress, in pounds per square inch of area of 
section; 

W =weight placed on a beam, in pounds; 
b = breadth of cross-section of beam, in inches; 
d = depth of beam (in.) = diam. of circ. section = altitude 
of triangular section = length of vertical side; 
e = amount of elongation or shortening, in inches; 

/= factor of safety; 

/ = length, in inches; 

p = pressure, in pounds per square inch; 

7r = ratio of circumference to diameter = 3.1416, nearly; 
g = constant used in formula for columns; 
r = radius of circular section; 

s = elastic set or deflection of a beam under a transverse 
(bending) stress, in inches; 
t = thickness of shell or hollow section. 

Tension, Compression, and Shear. —For tension, compres¬ 
sion (where the piece does not exceed 10 times its least diameter) 
and shear, 

AS 

P = - 

/ 

Breaking Stress. —To find the breaking stress, P, make 
/= 1. For safe load, take the values for/ and S from the accom- 


FACTORS OF SAFETY 


Material 

Steady 

Stress 

Varying 

Stress 

Shocks 

(Machines) 

Cast iron. 

6 

15 

20 

Wrought iron. 

4 

G 

10 

Steel. 

5 

7 

15 

Wood. 

8 

10 

15 

Brick and stone... . 

15 

25 

30 


panying tables of Factors of Safety and Ultimate Strengths 
espectively. 

















100 STRENGTH OF MATERIALS 


ULTIMATE STRENGTHS 


Material 

Tension 

Com¬ 

pression 

Shear 

Flexure 

Cast iron. 

20,000 

90,000 

20,000 

36,000 

Wrought iron. 

50,000 

50,000 

47,000 

50,000 

Steel. 

100,000 

150,000 

70,000 

120,000 

Wood. 

10,000 

8,000 

600 to 3,000 

9,000 

Stone. 


6,000 


2,000 

Brick. 

200 

2,500 




COEFFICIENT OF ELASTICITY 


Material 

Coefficient 
of Elasticity 

Elastic Limit 
for Tension 

Cast iron. 

15,000,000 

6,000 

Wrought iron. 

25,000,000 

25,000 

Steel. 

30,000,000 

50,000 

Wood. 

1,500,000 

3,000 


BENDING MOMENT AND DEFLECTION OF BEAMS 


Kind of Beam and Manner of Loading 


Cantilever, load at end. 

Cantilever, uniformly loaded. 

Simple beam, load at middle. 

Simple beam, uniformly loaded. 

Beam fixed at both ends, load at middle 

Beam fixed at both ends, uniformly 
loaded. 


Bending 

Moment 

Deflection 

M 


Wl 

x Wl* 

3 E I 

hwi 

, Wl* 
h E I 

i Wl 

, Wl* 

**E I 

\Wl 

. Wl* 
™*E I 

\Wl 

, Wl* 

hWl 

. Wl * 
r **E I 


















































STRENGTH OF MATERIALS 


101 


PROPERTIES OF VARIOUS SECTIONS 


Section 

I 

R 

G 2 

Solid 

wd 4 

Trd 3 


circular.. . iil?sli| 

d 2 


64 

32 

16 

Hollow 

circular.. . SHHIf 

7r(d 4 — di 4 ) 

64 

7r(d 4 —di 4 ) 

32 d 

d 2 +di 2 

16 


d 4 

d 3 

d 3 

Solid square 

Hollow Mf 

square '-" fed! 

Solid J §jj 

12 

d 4 — di 4 

12 

bd 3 

6 

d 4 — di 4 

6 d 

bd 3 

12 

d 2 +di 2 
~~ 12 

b 3 

rectangular J ||g 

12 

6 

12 

fH-r 

Hollow 1 |-^|X 

rectangular | 

bd 3 — b\d\ 3 

bd 3 — b\d\ 3 

b 3 d — bi 3 di 

12 

6d 

12(bd — bidi) 





Solid 

bd 3 

db 3 

d 3 

triangular JUflm 

36 

24 

18 

Soiid 

elliptic ... « pHI 

irbd 3 

irbd 3 

b 3 

1 ^P" 

64 

32 

16 

Hollow ["T^rT-w 

elliptic ... jr 

-£-(5d 3 -5idi 3 ) 

64 

ir(bd 3 — bidi 3 ) 

b 3 d — bi 3 di 

32 d 

16(6d —fcidi) 

f 

I-beam T 

bd 3 — bidi 3 

bd 3 — b\di 3 

5 3 d — 6i 3 di 

12 

6 d 

l'2(bd — bidi) 





Cross with 



d 3 

equal arms 



22.5 

Angle with 1 



d 2 

equal arms 1 



25 


















































102 


STRENGTH OF MATERIALS 


Elongation or Shortening Under Stress.—The amount of 
elongation or of shortening of a piece under a stress is given by 
the formula 

PI 

e = - 

AE 

The coefficient of elasticity E must be taken from the accom¬ 
panying table. 

Breaking Strength of a Beam.—To find the breaking strength 

of a beam, use the formula 

M = SR 


Obtain M and R from the accompanying tables according 
to the kind of beam and nature of cross-section. A simple 
beam is one merely supported at its ends. In the expression 
for R, d is always understood to be the vertical side or depth; 
hence, that beam is the stronger that always has its greatest 
depth or longest side vertical. The moment of inertia I is 
taken about an axis perpendicular to d, and lying in the same 
plane. 

The value of 5 for beams should be taken from the flexure 
column of table of Ultimate Strengths. 

Deflection in Beams Due to Loads.—To find the amount of 
deflection in a beam due to a load, substitute the values of 
W, l, E, and I in the different expressions for the deflection 5 
in the table Bending Moment and Deflection of Beams. The 
value of I is to be taken from the table Properties of Various 
Sections. 

Columns.—To find the breaking strength of a column, use 
the formula: 


P = 


SA 

£ 

1 +q— 

& 


The values of these quantities are taken from the accompany¬ 
ing tables. 

Ropes and Chains.—Let D = diameter of rope, in inches 
= diameter of iron from which link in chain is made; 

W= safe load, in tons of 2,000 lb. 

For common hemp rope, W= § D 2 . 




STRENGTH OF MATERIALS 


103 


For iron-wire rope, W =§ D 2 . 
For steel-wire rope, D 2 . 


CONSTANT USED IN FORMULA FOR COLUMNS 


Material 

Both Ends 
Flat or Fixed 

One End 
Round 

Both Ends 
Round 

Cast iron. 

1 

1.78 

4 

Wrought iron. 

5,000 

1 

5,000 

1.78 

5,000 

4 

Steel. 

36,000 

1 

36,000 

1.78 

36,000 

4 

Wood. 

25,000 

1 

25,000 

1.78 

25,000 

4 


3,000 

3,000 

3,000 


For close-link wrought-iron chain, W= 6 D~. 
For stud-link wrought-iron chain, W = 9 D 2 . 


WIRE ROPES 

Wire ropes for mine use are made of either iron or steel, and 
are generally round. Flat wire ropes are sometimes employed, 
but the round rope is the one generally used in American prac¬ 
tice, except in some of the deep metal mines having small com¬ 
partment shafts. Steel ropes are in most respects superior to 
iron ropes, and are therefore gaining favor every year. Their 
principal advantage is their greater strength; consequently, 
they can be made lighter and can pass around pulleys and drums 
with less injury than an iron rope of equal strength. 

Where great flexibility is required, such as in hoisting ropes, 
the strands are usually made up of 19 wires each, while haulage 
ropes have but 7 wires to the strand; yet, both kinds have 6 
strands. A hemp core is generally used, and in some cases a 
core is also placed in each strand, to further increase the flex¬ 
ibility of the rope. 





























104 


STRENGTH OF MATERIALS 


The lay of the rope is the twist or pitch of the wires in the 
strand, or of the strands in the rope. As the lay of the wires 
is less than that of the strands, each wire is exposed to external 
wear for short distances at intervals along the rope. 

In the ordinary lay, Fig. 1 (a), the wires are twisted in the 
opposite direction to the strands; this method prevents the 
rope from untwisting when in use, and the wires from unravel¬ 
ing when they are worn through or broken at the surface. 

In the Lang lay, view ( b ), the wires are twisted in the same 
direction as the strands, thus giving each wire a greater wearing 
surface, while the rope is smoother and will wear longer. After 



(a) (b) (c) 

Fig. 1 


the wires begin to break, unraveling becomes troublesome, and 
it is more difficult to splice a Lang-lay rope than an ordinary- 
lay one. Hoisting ropes, especially those used to raise and 
lower men, should not be spliced. 

The locked-wire rope, a cross-section of which is shown in 
(c), consists of wires of special cross-section formed in concen¬ 
tric layers. The lay of the inner wires is opposite to that of 

the outer ones, and somewhat 
longer. 

In fastening a rope to a drum, 
a great error is often made. Men 
who would not think of passing a 
rope around a pulley of too small 
diameter will insert it in the drum rim in such a way as to make 
a very sharp curve, as shown in Fig. 2 (a), and make a weak 
point in the rope that would not otherwise exist. The right 
way of passing the rope through the drum rim is shown in (6). 

Flattened-Strand Ropes.—Many ropes have flattened strands, 
as shown in Fig. 3; several wires thus take the wear of the rope 
instead of a single one, as is the case with a round strand when 
new. The manufacturers claim for these ropes longer life, more 



(«) (O 

Fig. 2 

















STRENGTH OF MATERIALS 


105 


uniform wear, greater flexibility, less liability of wires becom¬ 
ing brittle, and freedom from all tendency to spin or kink. It 



Fig. 3 


is also claimed that the smoother surface effects considerable 
saving in the wear of pulleys and sheaves. 


STRESS IN HOISTING ROPES ON INCLINED 

PLANES 


Rise per 
100 Ft. 
Hori¬ 
zontal 

Feet 

Angle 

of 

Incli- 
• nation 

Stress 
per Ton 
of 2,000 
Lb. 

Pounds 

Rise per 
100 Ft. 
Hori¬ 
zontal 

Feet 

Angle 

of 

Incli¬ 

nation 

Stress 
per Ton 
of 2,000 
Lb. 

Pounds 

5 

2° 52' 

140 

105 

46° 24' 

1,484 

10 

5° 43' 

240 

110 

47° 44' 

1,516 

15 

8° 32' 

336 

115 

49° 00' 

1.535 

20 

11° 10' 

432 

120 

50° 12' 

1,573 

25 

14° 03' 

527 

125 

51° 21' 

1,597 

30 

16° 42' 

613 

130 

52° 26' 

1.620 

35 

19° 18' 

700 

135 

53° 29' 

1,642 

40 

21°49' 

782 

140 

54° 28' 

1,663 

45 

24° 14' 

860 

145 

55° 25' 

1,682 

50 

26° 34' 

933 

150 

56° 19' 

1,699 

55 

28° 49' 

1,003 

155 

57°11' 

1,715 

60 

30° 58' 

1,067 

160 

58° 00' 

1,730 

65 

33° 02' 

1,128 

165 

58° 47' 

1,744 

70 

35° 00' 

1,185 

170 

59° 33' 

1,758 

75 

36° 53' 

1,238 

175 

60° 16' 

1,771 

80 

38° 40' 

1,287 

180 

60° 57' 

1,782 

85 

40° 22' 

1,332 

185 

61°37' 

1,794 

90 

42° 00' 

1,375 

190 

62° 15' 

1,804 

95 

43° 32' 

1,415 

195 

62° 52' 

1,813 

100 

45° 00' 

1,450 

200 

63° 27' 

1,822 


Wire-Rope Tables.—The accompanying wire-rope table is 
a rearrangement of the standard tables published in the 













106 


STRENGTH OF MATERIALS 


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cu 

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04 

& 

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t-H 

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CO 

a 

2 

H 

Q 

& 

c 

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-j 

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c 

M 

d 

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H 

CO 

«—I 

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W 


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CO OX OX ^-h ' — < • — • 

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cible Cast Steel 

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STRENGTH OF MATERIALS 


107 


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108 


STRENGTH OF MATERIALS 


catalogs of most of the American manufacturers of wire rope. 
The proper working load given in this table is one-fifth of the 
approximate breaking stress, that is, a factor of safety of 5 is 
used, and when the values given in this table are used this factor 
is supposed to allow for the bending stress. The sizes of sheaves 
or drums given in this table are largely empirical, but they are 
based on a long experience in the use of wire ropes, and in most 
cases represent the minimum diameter recommended by the 
rope makers. The factor of safety of 5 assumes ordinary con¬ 
ditions of working; where the conditions are extraordinary, and 
particularly in cases where men are to be hoisted, a larger factor 
than 5 is used, varying from 5 to 10. 

Stress in Hoisting Ropes on Inclined Planes.—The accom¬ 
panying table is based upon an allowance of 40 lb. per T. for 
rolling friction, but there will be an additional stress due to the 
weight of the rope and inclination of the plane. 


STARTING STRAIN ON HOISTING ROPES 


Experiment 

Strain in 
Rope 
Pounds 

Empty cage, lifted gently. 

4,030 

5,600 

8,950 

12.300 

11.300 
11,525 

19,025 

24,625 

26,850 

Empty cage, started with 25 in. of slack rope... 
Empty cage, started with 6 in. of slack rope... . 
Empty cage, started with 12 in. of slack rope... 
Cage and loaded cars, as weighed. 

Cage, and loaded cars, lifted slowly and gently. . 
Cage and loaded cars, started with 3 in. of slack 
rope. 

Cage and loaded cars, started with 6 in. of slack 
rope. 

Cage and loaded cars, started with 9 in. of slack 
rope. 



Starting Strain on Hoisting Rope.—In selecting a hoisting 

rope, due allowance must be made for the shock and extra 
strain imposed on the rope when the load is started from rest. 
Experiments made by placing a dynamometer between the 
rope and the cage have shown that the starting stress may be 
from two to three times the actual load. 














STRENGTH OF MATERIALS 


109 


The following table shows the results of a number of tests 
under different conditions, with slack chain, amount of load, 
and speed in starting. 


EXTRA STRAIN ON A HOISTING ROPE WITH A FEW 
INCHES OF SLACK CHAIN 


Dynamometer Tests 

Tons 

Hundred¬ 

weights 

First Test 

Empty cage lifted gently. 

Empty cage with 2\ in. slack chain. 

1 

16 

2 

10 

Empty cage with 6 in. slack chain. 

4 

0 

Empty cage with 12 in. slack chain. 

5 

10 

Second Test 

Cage and 4 empty cars weighed by 
machine. 

2 

17 

Cage and 4 empty cars lifted gently. 

3 

0 

Cage and 4 empty cars with 3 in. slack 
chain. 

5 

0 

Cage and 4 empty cars with 6 in. slack 
chain. 

5 

10 

Cage and 4 empty cars with 12 in. slack 

7 

0 

- t. 

Third Set 

Cage and full cars weighed by machine. . 

5 

1 

No. 1, Lifted gently. 

5 

1 

No. 2, lifted gently. 

5 

3 

No. 1, with 3 in. slack chain. 

8 

10 

No. 2, with 3 in. slack chain. 

8 

10 

No. 1, with 6 in. slack chain. 

10 

10 

No. 2 , with 6 in. slack chain. 

11 

10 

No. 1, with 9 in. slack chain. 

12 

10 

No. 2, with 9 in. slack chain. 

11 

10 


Sheaves.—To decrease the bending stresses the sheaves for 
wire-rope transmissions are generally of as large diameter as is 
practicable to give the required speed to the rope. Large 
sheaves are also advantageous because with them the rope is 
run at a high velocity allowing of a lower tension and permit¬ 
ting a rope of smaller diameter to be used than would be possible 
with smaller sheaves, provided, of course, that the span is 
of sufficient length to give the necessary weight. 






























110 


STRENGTH OF MATERIALS 


Sheaves are generally made of cast iron when not exceeding 
12 ft. in diameter, and when larger than this they are usually 
built up with wrought-iron arms. Sheaves, upon which the 
rope is to make but a single half-turn, are made with V-shaped 
grooves in their circumference. The bottom part of the groove 
is widened to receive the filling, which consists of some sub¬ 
stance to give a bed for the rope to run on and protect it from 
wear, and to increase the friction so that the rope will not slip. 
This filling is made of blocks of wood, rubber, leather, or other 
material Rubber and leather have been used separately, 
but blocks of rubber separated by pieces of leather have been 
found to give the best results. 

In the accompanying table, the ropes are made of cast steel 
and used on inclines. The ropes are composed of 6 strands 
of 7 wires each and have hemp cores. 


EFFECTS OF VARIOUS SIZE SHEAVES OR DRUMS ON 
LIFE OF WIRE ROPES 



Percentages of Life for Various Diameters 

Diameter 








of Rope 
Inches 

100 

90 

80 

75 

60 

50 

25 


Diameter of Sheaves or 

Drums, 

in Feet 

H 

16.00 

14.00 

12.00 

11.00 

9.00 

7.00 

4.75 

H 

14.00 

12.00 

10.00 

8.50 

7.00 

6.00 

4.50 

H 

12.00 

10.00 

8.00 

7.25 

6.00 

5.50 

4.25 

n 

10.00 

8.50 

7.75 

7.00 

6.00 

5.00 

4.00 

i 

8.50 

7.75 

6.75 

6.00 

5.00 

4.50 

3.75 

7 

5 

7.75 

7.00 

6.25 

5.75 

4.50 

3.75 

3.25 

3 

4 

7.00 

6.25 

5.50 

5.00 

4.25 

3.50 

2.75 

5 

8 

6.00 

5.25 

4.50 

4.00 

3.25 

3.00 

2.50 

1 

3 

5.00 

4.50 

4.00 

3.50 

2.75 

2.25 

1.75 


Wire-Rope Calculations.—The working load, also called the 
proper working load, is the maximum load that a rope should 
be permitted to support under working conditions. When a 
load is attached, the stress on a rope bending over a sheave, is 




















STRENGTH OF MATERIALS 


111 


WIRE AND SHEET-METAL GAUGES 


Gauge 

Number 

U. S. 
Stand¬ 
ard 
Sheet- 
Metal 
Gauge 

Inch 

British 

Im¬ 

perial 

Stand¬ 

ard 

Wire 

Gauge 

Mm. 

Birm¬ 

ingham 

Gauge 

Inch 

Amer¬ 
ican or 
Brown 
& 

Sharpe 

Gauge 

Inch 

Roeb- 

ling’s 

Gauge 

Inch 

Trenton 

Wire 

Co.’s 

Wire 

Gauge 

Inch 

0000000 

.5 

12.7 



.49 


000000 

.469 

11.78 



.46 


00000 

.438 

10.97 



.43 

.45 

0000 

.406 

10.16 

.454 

.46 

.393 

.40 

000 

.375 

9.45 

.425 

.40964 

.362 

.36 

00 

.344 

8.84 

.38 

.3648 

.331 

.33 

0 

.313 

8.23 

.34 

.32486 

.307 

.305 

1 

.281 

7.62 

.3 

.2893 

.283 

.285 

2 

.266 

7.01 

.284 

.25763 

.263 

.265 

3 

.25 

6.4 

.259 

.22942 

.244 

.245 

4 

.234 

5.89 

.238 

.20431 

.225 

.225 

5 

.219 

5.38 

.22 

.18194 

.207 

.205 

6 

.203 

4.88 

.203 

.16202 

.192 

.19 

7 

.188 

4.47 

.18 

.14428 

.177 

.175 

8 

.172 

4.06 

.165 

.12849 

.162 

.16 

9 

.156 

3.66 

.148 

.11443 

.148 

.145 

10 

.141 

3.26 

.134 

.10189 

.135 

.13 

11 

.125 

2.95 

.12 

.09074 

.12 

.1175 

12 

.109 

2.64 

.109 

.08081 

.105 

.105 

13 

.094 

2.34 

.095 

.07196 

.092 

.0925 

14 

.078 

2.03 

.083 

.06408 

.08 

.08 

15 

.07 

1.83 

.072 

.05707 

.072 

.07 

16 

.0625 

1.63 

.065 

.05082 

.063 

.061 

17 

.0563 

1.42 

.058 

.04526 

.054 

.0525 

18 

.05 

1.22 

.049 

.0403 

.047 

.045 

19 

.0438 

1.01 

.042 

.03589 

.041 

.04 

20 

.0375 

.91 

.035 

.03196 

.035 

.035 

22 

.0313 

.71 

.028 

.02535 

.028 

.028 

24 

.025 

.56 

.022 

.0201 

.023 

.0225 

26 

.0188 

.45 

.018 

.01594 

.018 

.018 

28 

.0156 

.38 

.014 

.01264 

.016 

.016 

30 

.0125 

.31 

.012 

.01002 

.014 

.014 

32 

.0101 

.27 

.009 

.00795 

.013 

.012 

34 

.0086 

.23 

.007 

.0063 

.01 

.01 

36 

.007 

.19 

.004 

.005 

.009 

.009 

38 

.0063 

.15 


.00396 

.008 

.008 

40 


.12 


.00314 

.007 

.007 




















112 


STRENGTH OF MATERIALS 


made up of two parts: (1) That due to the load on the rope, 
known as the load stress; (2) that due to the bending of the rope 
about a sheave or drum, known as the bending stress. That 
is, if 5 is the total safe stress, Si, is the bending stress, 5/ is the 
load stress, S = S/,+Si and Si = S — S&. The total stress must 
not equal the elastic limit of the material composing the rope 
and is usually taken as from one-third to one-fourth the approx¬ 
imate breaking stress. The proper size of rope to use to hoist 
a given weight may be taken directly from the tables just given, 
but these tables do not take account of the bending stress, 
except by allowing for it in the factor of safety assumed. 

The general formula for the bending stress is 


EaA 



in which Si, = bending stress; 

E = modulus of elasticity; 
a — diameter of each wire; 

D — diameter of drum or sheave; 

A — total area of wire cross-section, in inches. 

Wear of Wire Ropes.—The deterioration of wire ropes may 
be either external or internal, and may be due (1) to abrasion, 
due to the rubbing of the outside surface of the rope against 
other objects, or to the internal chafing of the wires composing 
the strands against one another; (2) to injury from overloading, 
to shock due to sudden starting of the load, or to repeated 
bendings about too sharp angles or over sheaves or rollers of too 
small a diameter for the size of the rope; (3) to rust or corrosion 
of the wire from acid waters, or to decay of the hemp cores. 

Lubrication of Ropes.—Mine water has a very corrosive 
action on wire ropes, and a rope will soon be destroyed unless 
the water is prevented from coming in contact with the metal 
of the rope. To avoid this corrosive action tar, black oil, or 
some lubricating preparation is applied to the rope, but any 
lubricant used must be free from acids or other substances that 
would corrode the wire. The lubricants to be used are generally 
specified by the manufacturers of the ropes, a list of which can 
be obtained from them when purchasing a rope. 




STRENGTH OF MATERIALS 


113 


Wire-Rope Fastenings. —An ordinary form of thimble-spliced 
fastening is shown in Fig. 1 (a). In this method, the wires, after 
being frayed out at the end and the rope bent around the 
thimble, are laid snugly about the main portion of the rope and 
securely fastened by wrapping with 
stout wire; the extreme ends that pro¬ 
ject below this wrapping are then folded 
back, as shown. Another style of 
thimble splicing is shown in ( b ). In 
this case the strands are interlocked 
as in splicing, and the joint is wrapped 
with wire as in the former method. 

The socket fastening is shown in (c ); 
the hole in which the rope end is fas¬ 
tened is conical in shape. The rope is 
generally secured by fraying out the 
interstices being filled up with spikes driven in tightly. The 
whole is finally cemented by pouring in molten Babbitt metal. 

A good fastening can be made if the wires, after being frayed 
out at the end, are bent upon themselves in hook fashion, the 
prongs of some being longer than others, so that the bunch will 
conform to the conical aperture of a socket, and the melted 
Babbitt metal finally run in as usual. This makes a much 
neater fastening than either of those shown in (a) and (b), 
but it does not possess nearly as much strength. The thimble 
possesses a serious disadvantage; it is usually made of a piece 
of curved metal bent around into an oval shape, as shown in 
(a) and (b), with the groove, in which the rope lies, outside, 
the ends coming together n a sharp point. When weight is 
placed on the rope, the strain on the thimble is apt to cause 
one end to wedge itself beyond or past the other, and with its 
sharp edge cut the strands in the splice. 

Splicing a Wire Rope. —To splice a wire rope the only tools 
needed are a cold cutter and hammer for cutting and trimming 
the strands, and two needles 12 in. long, made of good steel and 
tapered ovally to a point. Cut off the ends of the ropes to be 
spliced and unlay three adjacent strands of each back 15 ft.; 
cut out the hemp center to this point and relay the strands for 
7 ft. and cut them off. Pull the ropes by each other until they 



(a) 


Fig. 1 

wires at the end, the 






114 


STRENGTH OF MATERIALS 


have the position shown in Fig. 2 (a), cut off a and d', b and c' , 
view ( b ), making their lengths approximately 10 and 12$ ft., 
respectively, measured from the point where the hemp centers 
are cut. Place the ropes together, view ( b ); unlay e, d, c, view 
(a), keeping the strands together, and follow with e', d', d, 
view ( b ). Similarly, unlay /', a', b', view ( b ), and follow with 
/, a, b, until the rope appears as in view (c). Next run the 
strands into the middle of the rope. To do this, cut off the 
end of the strand e', view (c), so that when it is put in place 
it will just reach to the end x of the hemp core, and then push 



(c) 

Fig. 2 


one needle A through the rope from the under side, leaving 
two strands at the front of the needle, as shown. Push the 
other needle B through from the upper side and as close to the 
first needle A as possible, leaving the strands e and e' between 
them; place the first needle A on the knee and turn the other 
needle B around with the coil of the rope, and force the strand 
e' into the center of the rope. Repeat this operation with the 
other ends and cut them off so that the ends coming together 
in the center of the rope will butt against each other as nearly 
as possible. 





























STRENGTH OF MATERIALS 


115 


Ordinary Long Splice.—The tools required to make a long 
splice in wire rope are a pair wire nippers, for cutting off strands; 
a pair pliers, for pulling through and straightening ends of 
strands; two marline-spikes, one round and one oval, for open¬ 
ing strands; a knife to cut hemp center; two clamps, to untwist 
rope to insert ends of strands, or, in place of them, two short 
hemp-rope slings, with a stick for each as a lever; a wooden 
mallet, and some rope twine. Also, a bench and vise are handy. 
The length of the splice depends on the size of the rope. The 
larger ropes require the longer splices. The splice of ropes 
from f in. to g in. in diameter should not be less than 20 ft.; 
from J in. to lg in., 30 ft.; and from If in. up, 40 ft. 



Fig. 3 


To splice a rope, tie each end with a piece of cord at a distance 
equal to one-half the length of the splice, or 10 ft. back from 
the end, for a f in. rope, after which unlay each end as far as the 
cord. Then cut out the hemp center, and bring the two ends 
together as close as possible, placing the strands of the one end 
between those of the other, as shown in Fig. 3 (a). Remove 
the cord k from one end M of the rope, and unlay any strand a, 
and follow it up with the strand of the other end M' of the rope 
that corresponds to it, as a !. Leave out about 6 in. of a and 
cut off a' about 6 in. from the rope, thus leaving two short ends, 
as shown at P in view ( b ), which must be tied for the present 








116 


STRENGTH OF MATERIALS 


by cords as shown. The cord k should again be wound around 
one end M of the rope, view (a), to prevent the unraveling of 
the strands. Remove the cord k' on the other end M' of the 
rope, and unlay the strand b; follow it up with the strand b', 
leaving out the ends and tying them down, as at view ( b ). 
Replace the cord k' for the same purpose as stated before. 
Again remove the cord k and unlay the next strand c, view (a), 
and follow it up with c', stopping, however, this time within 
4 ft. of the first set. Continue this operation with the remaining 
6 strands, stopping 4 ft. short of the preceding set each time. 
The strands are now in their proper places, with the ends pass¬ 
ing each other at intervals of 4 ft., as shown in view (c). To 
dispose of the loose ends, clamp the rope in a vise at the left of 
the strands a and a', and fasten a clamp to the rope at the right 
of these strands; then remove the cords tied around the rope 
that holds these two strands down; after which turn the clamp 
in the opposite direction to which the rope is twisted, thereby 
untwisting the rope, as shown in view ( d ). The rope should be 
untwisted enough to allow its hemp core to be pulled out with 
a pair of nippers. Cut off 24 in. of the hemp core, 12 in. at 
each side from the point of intersection of the strands a and a', 
and push the ends of the strands in their place, as shown. 
Then allow the rope to twist up to its natural shape, and remove 
the clamps. After the rope has been allowed to twist up, the 
strands tucked in generally bulge out somewhat. This bulging 
may be reduced by lightly tapping the bulged part of the strands 
with a wooden mallet, which will force their ends farther into 
the rope. 


CHAINS 

The links of iron chains are usually made as short as is con¬ 
sistent with easy play, so as to make them less liable to kink, 
and also to prevent bending when wound around drums, sheaves, 
etc. The weight of close-link chain is about 3 times the weight 
of the bar from which it is made, for equal lengths. 

The strength of a chain link is less than twice that of a straight 
bar of a sectional area equal to that of one side of the link. A 
weld exists at one end and a bend at the other, each requiring 
at least one heat, which produces a decrease in the strength. 



STRENGTH OF MATERIALS 


117 


The report of the committee of the U. S. Testing Board, on 
tests of wrought-iron and chain cables, is shown in the accom¬ 
panying table. 


ULTIMATE RESISTANCE AND PROOF TESTS OF 
WROUGHT-IRON CHAIN CABLES 


Diam. 
of Bar 

Inches 

Average 
Resist. = 
163% of 
Bar 

Pounds 

Proof 

Test 

Pounds 

Diam. 
of Bar 

Inches 

Average 
Resist. = 
163% of 
Bar 

Pounds 

Proof 

Test 

Pounds 

1 

71,172 

33,840 

1 16 

162,283 

77,159 

liV 

79,544 

37,820 

If 

174,475 

82,956 

n 

88,445 

42,053 

1 16 

187,075 

88,947 


97,731 

46,468 

1 4 

200,074 

95,128 

n 

107,440 

51,084 

113 . 

-*-16 

213,475 

101,499 

1 - 5 . 

1 16 

117,577 

55,903 

1 7 

1 8 

227,271 

108,058 

It 

128,129 

60,920 

1 15 

1 16 

241,463 

114,806 

1 A 

139,103 

66,138 

2 

256,040 

121,737 

U 

150,485 

71,550 





HORSEPOWER OF MANILA ROPES 


Diam. of 
Rope. In. 

Weight per 
Foot. Lb. 

Breaking 

Strain 

Lb. 

Working 
Strain. Lb. 

1,000 Ft. 
per Min. 

2,000 Ft. 
per Min. 

3,000 Ft. 
per Min. 

H. P. 

Tens. 

Wt. 

CL 

W 

Tens. 

Wt. 

CL 

M 

Tens. 

Wt. 

5 

.15 

4,000 

121 

2i 

90 

41 

90 

61 

80 

3 

.18 

5,000 

151 

2 f 

110 

51 

110 

7 f 

100 

7 

.27 

7,500 

227 

41 

170 

81 

170 

Ilf 

160 

1 

.33 

9,000 

272 

5 

200 

10 

200 

14 

180 

H 

.45 

12,250 

371 

7 

280 

131 

270 

19 

250 

n 

.50 

14,000 

424 

8 

320 

151 

310 

22 

290 

1 3 

.65 

18.062 

547 

101 

410 

20 

400 

28f 

370 

11 

.73 

20,250 

613 

1H 

460 

22 

440 

311 

420 

if 

.82 

25,000 

760 

141 

570 

27 f 

550 

391 

520 

if 

1.08 

30,250 

916 

17 

680 

331 

660 

471 

630 

2 

1.27 

36,000 

1,000 

201 

810 

40 

790 

561 

740 


5,000 Ft. 
per Min. 


cl 


CO 

C 



V 

H 


10 i 
16 
19 
26 
29 5 
38 5 
431 
551 
64 f 
771 


70 

90 

130 

150 

210 

240 

310 

350 

448 

520 

620 


Wt. 











































118 


H YDROMECHA NICS 


HYDROMECHANICS 


HYDROSTATICS 

Hydrostatics treats of liquids at rest under the action of 
forces. If a liquid is acted on by a pressure, the pressure per 
unit of area exerted anywhere on the mass of liquid is trans¬ 
mitted undiminished in all directions, and acts with the 
same force on all surfaces, in a direction at right angles to 
those surfaces. 

Downward Pressure of Liquids.—The pressure on the bot¬ 
tom of a vessel containing a liquid is independent of the shape of 
the vessel, and is equal to the weight of a prism of the liquid 
whose base is the same as the bottom of the vessel, and whose 
altitude is the distance between the bottom and the upper sur¬ 
face of the liquid, plus the pressure per unit of area upon the 
upper surface of the liquid multiplied by the area of the bottom 
of the vessel. 

Upward Pressure of Liquids.—The upward pressure on any 
submerged horizontal surface equals the weight of a prism of the 
liquid whose base has an area equal to the area of the submerged 
surface, and whose altitude is the distance between the sub¬ 
merged surface and the upper surface of the liquid, plus the 
pressure per unit of area on the upper surface of the liquid mul¬ 
tiplied by the area of the submerged surface. 

Lateral Pressure of Liquids.—The pressure on any vertical 
surface due to the weight of the liquid is equal to the weight of 
a prism of the liquid whose base has the same area as the ver¬ 
tical surface, and whose altitude is the depth of the center of 
gravity of the vertical surface below the level of the liquid. 
Any additional pressure is to be added, as in the previous cases. 

Pressure of Liquids on Oblique Surfaces.—The pressure 
exerted by a liquid in any direction on a plane surface is equal 
to the weight of a prism of the liquid whose base is the projection 
of the surface at right angles to the given direction, and whose 
height is the depth of the center of gravity of the surface below 
the level of the liquid. 



H YDROMECHA NICS 


119 


If a cylinder is filled with water, and a pressure applied, 
the total pressure on any half section of the cylinder is equal 
to the projected area of the half cylinder (or diameter length of 
cylinder) multiplied by the depth of the center of gravity of the 
half cylinder, multiplied by the weight of 1 cu. in. of water, 
plus the diameter of the shell, multiplied by the pressure per 
square inch, multiplied by the length of the cylinder. 

If d = diameter, in inches, and Z = length of cylinder, in inches, 
the pressure due to the weight of the water when the cylinder 


l 

is vertical upon the half cylinder = d X l X - X weight of 1 cu. in. 

Z* 2 

of water = dX~X weight of 1 cu. in. of water. 


The pressure, in pounds per square inch, due to a head of 
water is equal to the head in feet multiplied by .434. The head 
equals the pressure, in pounds per square inch, multiplied by 
2.304. 


Flow of Water Through Pipes.—The following formulas for 
the flow of water through pipes are those arranged by Gould, 
in which 


Q — amount of water, in cubic feet per second; 
g = U. S. gallons per minute; 

D = diameter of pipe, in feet; 
d = diameter of pipe, in inches; 

H = total head, in feet; 
h = head per 1,000 ft.; 

V = velocity, in feet per second. 

Pipes above 8 in. in diameter, rough inside surface. 

<2 = \lD b h = D*^Dh; V=1.27 ^Dh 


For diameter, in inches, 


Q = 


288 



/ 


Pipes between 3 and 8 in. in diameter, rough inside surface, 
Q = 0.89 Vd 5 /j = 0.89 D»^Dh; V = 1.13 ^Dh 
Large pipes, smooth inside surface, 

<2 = 1.4 = 1.4 D*^Dh; V = 1.78^ Dh 

Small pipes, smooth inside surface, 

Q = 0.89 ‘^2D b h = 1.25 dWdA; V = 1.6 VdA 



120 


HYDROMECHANICS 


It is best to calculate any pipe line by the formula for pipes 
having a rough internal surface, because all pipes become more 
or less rough with use. 

Siphons.—When any part of the pipe line rises above the 
source of supply, such a line is called a siphon. If this rise 
is greater than the height of the water barometer (34 ft. at 
sea level), water will not flow through the siphon. The flow 
through the siphon will be the same as that through any pipe 
line so long as there is no accumulation of air at the highest 
point of the line; but such an accumulation will decrease or 
entirely stop the flow. All siphons should be provided at their 
highest points with valves for discharging the air and intro¬ 
ducing water to fill the siphon, and it is usually best to trap the 
lower end of the pipe so that air cannot enter it, and to enlarge 
the upper end so as to reduce the loss of the stream in entering. 
For a siphon to work well, the fall between the intake and the 
discharge end should be considerable, if the rise amounts to 
much. 


DAMS 

Construction of Dams in Mines.—Dams may be constructed 
in mines, either to isolate a portion of the workings so that it 



can be flooded to extinguish fires, or, where an extremely wet 
formation has been penetrated, to prevent the water from 











HYDROMECHANICS 


121 


flowing into the workings. But in either case the dam should 
have sufficient strength to resist any column of water that will 
come against it. The dam should be arched toward the direc¬ 
tion from which the pressure comes, and should be given a good 
firm bearing in both walls and in the floor and roof. A brick 
dam that was constructed to isolate a portion of the seam so 
that it might be flooded to extinguish a mine fire is shown in 
Fig. 1, (a) being the plan and ( b ) a cross-section. This dam 
is composed of three brick arches, each 5 ft. thick, placed one 
against the other so that they act as one solid structure. The 
gangway at this point is about 20 ft. wide, and the distance 
to the next upper level is about 119 ft. It was intended that 
this should be the maximum head of water that the dam would 
have to resist, though it was made sufficiently strong to resist 
a head of water reaching the surface. The separate walls were 
constructed one at a time, and the cement allowed to set before 
the next wall was placed. The back wall was carried to a 
greater depth and height than the others, so as to make sure 
that all slips or partings had been closed. The total pressure 
upon the dam when the water was in the mine was about 
70,000 lb. per sq. ft. 

Dams constructed to permit the flooding of a mine usually 
require no passages through them, but dams constructed to 
confine the water to certain parts of the workings, and so 
reduce pumping charges, usually have both manways and drain 
pipes through them. Fig. 2 (a) shows the plan and ( b ) shows 
the cross-section of such a dam constructed to keep the water 
that came from some exploring drifts out of the mine work¬ 
ings. As originally constructed, it consisted of a sandstone 
dam 10 ft. thick and arched on the back face with a radius of 
6 ft. A piece of 20-in. pipe provided a manway through the 
masonry and was held in place by three sets of clamps and 
bolts passing through the stone work. A 5-in. drain pipe was 
also carried through the dam and secured by clamps. When 
the pressure came upon the dam it was found to leak, so the 
water was drained off and a 22-in. brick wall built 2 ft. 4 in. 
back of the dam, the space between being filled with concrete, 
and the manway and drain pipe extended through the brick 
wall. Before closing the drain pipe, horse manure was fastened 


122 


HYDROMECHANICS 


against the face of the brick wall by means of a plank par¬ 
tition. After this the manway and drain pipe were closed, 
and when the pressure came on, the dam was found to leak a 




Fig. 2 

small amount, but this soon practically ceased, showing that 
the manure had closed the leaks. A gauge in the head of the 
manway on this dam showed a pressure of 211 lb., which cor¬ 
responded to a static head of 640 ft. of water. The total 
















































































H YDROMECHA NICS 


123 


pressure against the dam was something over 800 T., which it 
successfully resisted. 

Reinforced concrete is largely used for mine dams, as these 
dams are strong, impermeable, quickly made, and of reason¬ 
able cost. Old rails, strap iron, etc. may be used for the rein¬ 
forcing material. 

In mine work, dams are used for retaining water in reser¬ 
voirs, for diverting streams in placer mining, and for storing 
debris coming from placer mines in canons or ravines. 

Foundations for dams must be solid to prevent settling, 
and water-tight to prevent leakage under the base of the dam; 
whenever possible, the foundation should be solid rock. Gravel 
is better than earth, but when gravel is used sheet piling must 
be driven under the upper toe of the dam, to prevent water 
from seeping through the formation under the dam. Vegetable 
soil should be avoided, and all porous material, such as sand, 
gravel, etc. should be stripped off until hard pan or solid rock 
is reached. 

Abutments are timber, masonry, or dry stonework struc¬ 
tures at the ends of a dam. If possible, they should have a 
curved outline, and should be so placed that there is no pos¬ 
sibility of the water overflowing them, or getting behind them 
during floods. 

If the regular discharge from a dam tak'es place from the 
main face, the discharge gates may be arranged in connection 
with one of the abutments, or by means of a tunnel and cul¬ 
vert through the dam. In either case, some structure should 
be constructed above the outlet so as to prevent driftwood, 
brush, and other material from stopping the gates. When 
the discharge gates are placed at one side of the dam, they are 
usually arranged outside of the regular abutment, between it 
and another special abutment, the discharge being through a 
series of gates into a flume, ditch, or pipe. 

Spillways or waste ways, are openings provided in a dam for 
the discharge of water during floods or freshets, or for the dis¬ 
charge of a portion not being used at any time. The spillway 
may be over the crest of the dam, or, where the topography 
favors such a construction, the main dam may be of sufficient 
height to prevent water from ever passing its crest and the 





124 


H YDROMECHA NICS 


spillway arranged at another outlet over a lower dam. Waste 
ways, proper, are openings through the dam, to provide for 
the discharge of the large quantities of water that come down 
during freshets or floods. 

Wooden Dams.—Wooden dams are constructed of round, 
sawed, or hewn logs. The timbers are usually at least 1 ft. 
square, or, if round, from 18 to 24 in. in diameter. A series 
of cribs from 8 to 10 ft. square are constructed by building up 
the logs log-house fashion and securing them together with 
treenails. The individual cribs are secured to one another 
with treenails or by means of bolts. The cribs are usually 
filled with loose rock to keep them in place, and in many cases 
are secured to the foundation by means of bolts. The dam is 
made water-tight by a layer of planking on the upper face and 
if the spillway is over the crest of the dam it will be necessary 
to plank the top of the cribs, and, in most cases, to provide 
an apron for the water to fall on. The apron may be set on 
small cribs, or on timbers projecting from the cribs of the 
dam itself. 

Stone Dams.—Where cement or lime is expensive, and suit¬ 
able rubble stone can be obtained, dams are frequently con¬ 
structed without the use of mortar. The upper and lower faces 
of the dam should be of hammer-dressed stone, carefully 
bonded; sometimes the stones in the lower face of the dam are 
anchored by means of bolts. The dam can be made water¬ 
tight by means of a skin of planking on the upper face. In 
case water should pass over the crest of such a dam, much of it 
would settle through the openings in the stone into the interior 
of the dam, and thus subject the stones in the lower portion 
of the face to a hydrostatic pressure; for this reason, culverts 
or openings should be made through the lower portion of the 
dam, to discharge any such water. When such dams as this 
are constructed, the regular spillway is not placed over the face 
of the dam, but at some other point, and usually over a timber 
dam. 

Earth Dams.—Earth dams are used for reservoirs of mod¬ 
erate height. They should be at least 10 ft. wide on top, a 
height of more than 60 ft. being unusual. When the material of 
which the dam is composed is not water-tight, it is sometimes 


H YDROMECHA NICS 


125 


necessary to construct, in the center of the regular dam, a 
narrow dam of clay mixed with a certain proportion of sand. 
This puddle wall should not be less than from 6 to 8 ft. thick 
at the top and should have a slight batter on each side. It is 
constructed during the building of the dam, and should be 
protected from contact with the water by a considerable thick¬ 
ness of earth on the upper face. The upper face of an earthen 
dam is frequently protected by means of plank or a pavement 
of stone. The lower face is frequently protected by means 
of sod, or sod and willow trees. Sometimes earth dams are 
provided with a masonry core in place of the puddle wall, 
to render them water-tight. 

Masonry Dams.—High masonry dams should always be 
designed by a competent hydraulic engineer. Masonry dams 
are not, as a rule, used for hydraulic mining, as the length of 
time during which the dam is required rarely warrants the 
expense of the construction of a masonry dam. 

Debris Dams.—Dams or obstructions are sometimes placed 
across the bed of the stream to hold back culm, etc., from mines, 
and to prevent damage to the valleys below. They are made 
of stone, timber, or brush. No attempt is made to render 
these debris dams water-tight, as their only object is to retard 
the flow of the stream and to give it greater breadth of dis¬ 
charge, so that the water naturally drops and deposits the 
sediment that it is carrying. The sediment soon silts or fills 
up against the face of the dam, the area above the dam becom¬ 
ing a flat expanse or plain over which the water finds its way 
to the dam. 


RESERVOIRS 

In selecting a site for a reservoir, the points to be observed 
are: 

A proper elevation above the point at which the water is 
required. 

The total supply available, including observations as to the 
rainfall and snow fall. 

The formation and character of the ground, with reference 
to the amount of absorption and evaporation. 



128 


H YDROMECHA NICS 


PUMP MACHINERY 

Pumps are used for unwatering mines, handling water at 
placer mines, irrigation, water-supply systems, boiler feeds, 
etc. For unwatering mines, two general systems of pumping 
are used: In one, the pump is placed in the mine and is oper¬ 
ated by a motor on the surface, the power being transmitted 
through a line of moving rods; in the other, both the motor 
and the pump are placed in the mine, the motor being an engine 
driven by steam, compressed air, hydraulic motor, or an elec¬ 
tric motor. 

Cornish Pumps.—Any method of operating pumps by rods 
is commonly called a Cornish system. This system requires 
no steam line down the shaft and is independent of the depth 
of water in the mine, so that the pump is not stopped by the 
drowning of a mine, but the moving rods are a great incon¬ 
venience in the shaft, and they absorb much of the power 
through friction. 

Simple and Duplex Pumps.—In the simple pump, a steam 
cylinder is connected directly to a water cylinder, and the 
steam valves are operated by tappets. Such a pump is more 
or less dependent on inertia at certain points of the stroke to 
insure the motion of the valves, hence will not start from any 
place and is liable to become stalled at times. In the duplex 
pump, two steam cylinders and two water cylinders are arranged 
side by side, and the valves are so placed that when one piston 
is at mid-stroke it throws the steam valve for the other cylin¬ 
der, etc. With this arrangement, the pump will start from 
any point, and can never be stalled for lack of steam, due to the 
position of the valves. Ordinarily, duplex pumps are to be 
preferred for mine work. 

The packing for the water piston of a pump may be either 
inside or outside. As a rule, inside-packed pumps should be 
avoided in mines, because acid or gritty waters are liable to 
cut the packing and make the pumps leak in a very short time. 
For dipping work in single stopes or entries, small single or 
duplex outside-packed pumps may be used. It is generally best 
to operate such pumps by compressed air, for the exhaust will 
then be beneficial to the mine air. If steam is used, it is 


HYDROMECHANICS 


127 


frequently necessary to introduce a trap and remove entrained 
water from the steam before it enters the pump, and dispose 
of the exhaust by piping it out or condensing it. Such isolated 
steam pumps are about the most wasteful form of steam-driven 
motor in existence. 

For sinking, center-packed single or duplex pumps are usu¬ 
ally employed, the duplex style being the better. For station 
work, where much water is to be handled, large compound, 
or triple-expansion, condensing, duplex pumping engines are 
employed. They may, or may not, be provided with cranks 
and a flywheel; engineers differ greatly upon this point, but as a 
rule, for very high lifts and great pressures, the flywheel is used. 

Capacity of Pumps and Horsepower Required to Raise 
Water.—To find the capacity of pumps and the horsepower 
required to raise water any distance, 

Let Q = cubic feet of water per minute; 

G = U. S. gallons per minute; 

G' = U. S. gallons per hour; 

d = diameter of cylinder, in inches; 

1 = stroke of piston, in inches; 

N = number of single strokes per minute; 

v = speed of piston, in feet per minute; 

W = weight moved, in pounds per minute; 

P — pressure, in pounds per square feet = 62.5 XH; 

p = pressure, in pounds per square inch = .433XH; 

H = height of lift, in feet; 

H. P. = horsepower. 

7 r dT- IN 

Then, Q = ~X -X~ = .0004545 NdH 

4 144 12 

7 T NdH 

G = -X -=.0034 NdH. G' = .20iNdH 

4 231 


The diameter of piston required for a given capacity per 
minute will be 

d = 46.9\/—= 17.15\/^;, or d= 13.54\/- = 4.95 

\ Nl \ Nl \ v \ v 

The actual capacity of a pump will vary from 60% to 95% 
of the theoretical capacity, depending on the tightness of the 
piston, valves, suction pipe, etc. 



128 


HYDROMECHANICS 


QP QHX 144 X .433 QH Gp 

H. P.= —-— =-=-=- 

33.0C0 33,000 529.2 1,714.5 

The actual horsepower required will be considerably greater 
than the theoretical, on account of the friction in the pump; 
hence, at least 20% should be added to the power for friction 
and usually about 50% more is added to cover leaks, etc., 
so that the actual horsepower required by the pump is about 
70% more than the theoretical. 

Limit of Suction.—Theoretically, a perfect pump will raise 
water to a height of nearly 34 ft. at the sea level; but owing to 
the fact that a perfect vacuum can never be attained with the 
pump, that the water always contains more or less air, and that 
more or less watery vapor will form below the piston, it is never 
possible to reach this theoretical limit, and, in practice, it is 
not possible to draw water much, if any, over 30 ft. at the sea 
level, even when the water is cold. Warm water cannot be 
lifted as high as cold water, because a larger amount of watery 
vapor forms. With boiler feed-pumps handling hot water, 
the water should flow to the pumps by gravity. 

Power Pumps.—Where comparatively small amounts of 
water are to be handled and power is available, belt-driven 
power pumps are very much more efficient than small steam 
pumps. Where water is to be delivered from isolated work¬ 
ings to the sumps for the large station pumps, electrically 
driven power pumps are far more efficient than steam pumps. 

Miscellaneous Forms of Water Elevators.—In the jet pump 
the energy of the jet of water is utilized for raising a larger 
volume through a small distance, or a. mixture of water and solid 
material through a short distance. 

The pulsometer consists of two chambers in a large casting, 
with suitable automatic valves arranged at the top and bot¬ 
tom of the chambers. Steam is introduced into one of the 
chambers, then the valve at the top closed. As this steam 
condenses, it forms a vacuum that draws water from the suc¬ 
tion into the chamber. When the chamber is filled with 
water, steam is again introduced and forces the water out 
through the discharge pipe. The operation is then repeated, 
more water being drawn in by the condensation of the 
steam. 






H YDROMECHA NICS 


129 


Air-lift pumps have not been successful as mine pumps, 
owing to the ratio between the part immersed and the lift. 

In centrifugal pumps, the height of lift depends on the tan¬ 
gential velocity of the revolving disk of pump and the quan¬ 
tity of water discharged, and is proportional to the area of the 
discharge orifices at the circumference of the disk. The most 
efficient total lift for the centrifugal pump is, approximately, 
17 ft., and for small lifts the centrifugal pump is much more 
efficient than any style of piston pump. For a given lift, the 
total efficiency of a centrifugal pump increases with the size 
of the pump. Centrifugal pumps are always designated by 
the size of their outlet, as, for instance, a 2-in. or 4-in. pump, 
meaning a 2-in. or 4-in. discharge pipe. Under the most favor¬ 
able circumstances, the efficiency of the centrifugal pump may 
be practically 70%; that is, the pump may do an amount of 
work upon the water that is theoretically equal to 70% of 
the power furnished to the pump. Pumping engines working 
against high heads, and operated by the most improved class 
of engines, may attain an efficiency of practically 85%. 

Where only a limited amount of water collects in the mine 
workings, it is frequently removed by means of a special water 
bucket or water car during the hours that the hoisting engine 
would otherwise be idle. Where very large amounts of water 
are to be removed, it 
has also been found 
economical to remove 
them by means of 
special water buckets; 
especially in the case 
of deep shafts. 

Pumps for Acid 
Waters.—Where mine 
waters are acid in their 
nature, brass or brass- 
lined pumps are usu¬ 
ally employed. The 



Fig. 2 


Fig. 1 

pipes for such pumps should be of brass or copper tubing, 
or should be lined with some substance that will not be affected 
by the acid of the water. Sometimes wooden linings are 






130 


STEAM 


employed, placed as shown in Fig. 1, which is a section of 
the pipe with the lining complete. In Fig. 2 is shown a cross- 
section of one of the individual boards used in the lining. 
These boards are usually made of pine about f in. thick, and 
are grooved on each end as shown. They are sprung in so as 
to complete a circle on the inside of the pipe, and then long, 
thin, wooden keys are driven into the grooves. When the 
water is allowed to go into the pipes, the linings swell and make 
all joints perfectly tight. Elbows and other crooked sections 
are lined with sheet lead beaten in with a mallet. 


STEAM 


FUELS 

Classification of Coals.—Coals may be broadly divided into 
two classes: anthracite, or hard, coal, and bituminous, or soft, 
coal. The subdivisions given, however, are entirely arbitrary, 
as the different varieties of coal are found to shade insensibly 
into one another. 

Anthracite, or hard coal, which has a specific gravity of 1.30 
to 1.70, is the densest, hardest, and most lustrous of all varieties. 
It burns with little flame and no smoke, but gives a great heat, 
and contains very little volatile combustible matter. Its color 
is deep black and shining; sometimes it is iridescent; its frac¬ 
ture is conchoidal. Semianthracite coal is not so dense nor so 
hard as the true anthracite; its percentage of volatile combus¬ 
tible matter is somewhat greater, and it ignites more readily. 

Bituminous, or soft, coal, which has a specific gravity of 1.25 
to 1.40, is generally brittle. It has a bright pitchy or greasy 
luster, and is rather fragile as compared with anthracite. It 
burns with a yellow smoky flame, and gives, on distillation, 
hydrocarbon oils or tar. Under the term bituminous are 
included a number of varieties of coal that differ materially 
under the action of heat, giving rise to the general classification 
Coking or caking coals, and free-burning coals. Semibituminous 




STEAM 


131 


coal has the same general characteristics as the bituminous, 
although it is usually not so hard, and its fracture is more 
cuboid. The percentage of volatile combustible matter is less. 
It kindles readily, burns quickly with a steady fire, and is much 
valued as a steam coal. 

Coking coals are those that become pasty or semiviscid in the 
fire; and, when heated in a closed vessel, become partially fused 
and agglomerate into a mass of coherent coke. This property 
of coking may, however, become greatly impaired, if, indeed, 
not entirely destroyed, by weathering. Free-burning coals have 
the same general characteristics as the coking coals, but they 
burn freely without softening, and do not fuse or cake together 
in any sensible degree. 

Splint coal has a dull black color, and is much harder and 
less frangible than the coking coal. It is readily fissile, like 
slate, but breaks with difficulty on cross-fracture. It ignites 
less readily, but makes a hot fire, constituting a good house 
coal. 

Cannel coal differs from the ordinary bituminous coal in its 
texture. It is compact, with little or no luster and without any 
appearance of a banded structure. It breaks with a smooth 
conchoidal fracture, kindles readily, and burns with a dense 
smoky flame. It is rich in volatile matter, and makes an 
excellent gas coal. Its color is dull black and grayish-black. 

Lignite, or brown coal, often has a lamellar or woody structure; 
is sometimes pitch black, but more often rather dull and brown¬ 
ish black. It kindles readily and burns rather freely with a 
yellow flame and comparatively little smoke, but it gives only a 
moderate heat. It is generally non-coking. The percentage 
of moisture present is invariably high—from 10% to 30%. 

Composition of Coals.—A proximate analysis determines 
the proportion of those products of a coal having the most 
important bearing on its uses. These substances as usually 
presented are: moisture, or water, volatile combustible matter, 
fixed carbon, sulphur, and ash. In addition to these, the fol¬ 
lowing physical properties are generally given: color of ash, 
specific gravity and strength or hardness. The determination 
of these eight factors gives a fair general idea of the adapta¬ 
bilities of a coal. 


132 


STEAM 


Moisture, or water, in coal, has no fuel value. 

Volatile combustible matter is an important constituent of 
coal, the amount and quality deciding whether a coal is suitable 
for the manufacture of illuminating gas. The coking of coal 
also is largely dependent on this constituent. 

The fixed carbon is the principal combustible constituent in 
coal, and, in bituminous and semibituminous coals, the steaming 
value is in proportion to the percentage of fixed carbon. 

Sulphur will burn and develop heat and is not inert like 
moisture and ash; but it corrodes grates and boilers. In the 
blast furnace, it injures iron, and produces a hot short pig, and 
is objectionable in coal for forge use. For gas making, the 
sulphur must be removed. 

Ash is an inert constituent, which means that 20 lb. of weight 
must be handled and 20 lb. loss per T. of coal for each per cent, 
of ash present. The color of the ash furnishes a rough estimate 
of the amount of iron contained in a fuel. Iron in an ash makes 
it more fusible, and incieases its tendency to clinker. 

The specific gravity is an important factor when there is 
restriction of space, as on railway cars and in ship bunkers. 
A given bulk of anthracite coal will weigh from 10% to 15% 
more than the same bulk of bituminous coal, so that from 10% 
to 15% more pounds of fuel can be carried in the same place. 

Strength or hardness is valuable in preventing waste. A very 
soft coal is shipped in lump. Strength is a requisite for the use 
of raw coal in the blast furnace, and also to prevent excessive 
loss of coal through the grates in ordinary furnaces. 

Coke is the fixed carbon of a coal, a fused and porous product 
produced by the distillation of the gaseous constituent. For 
metallurgical use, it should be firm, tough, and bright, with a 
sonorous ring, and should contain not over 1% of sulphur. 
For blast-furnace use, a dense coke is objectionable, and the 
best is the one with the largest cell structure and the hardest 
cell wall. A high percentage of volatile hydrocarbon is, as a 
rule, necessary for a good coking coal. 

The essentials of a good gas coal are a low percentage of ash, 
say 5%, and of sulphur, say £ of 1%, a generous share, say 
37% to 40% of volatile matter, charged with rich illuminating 
hydrocarbons. It should yield, under present retort practice, 


STEAM 


133 


85 candle-feet to the pound carbonized. It should be suffi¬ 
ciently dense to bear transportation well, so that, when carried 
long distances, it will not arrive at its destination largely 
reduced to slack or fine coal of the consistency of sand; and it 
should possess coking qualities that will bring from the retorts, 
after carbonization, about 60% of clean, strong, bright coke. 

A good coal for blacksmith purposes should have a high heating 
power, should contain a very small amount of sulphur, if any, 
should coke sufficiently to form an arch on the forge, and should 
also be low in ash. 

The analysis of a coal does not necessarily determine its 
value or the uses to which it can be put. However, for examin¬ 
ing the analyses given in the accompanying tables, certain 
standards may be adopted as showing in a general way about 
what the analysis of coal should be for certain purposes. For 
steam purposes, the semibituminous coals have established 
reputations; for gas coals, that from Youghiogheny, Pa., is well 
known; for blacksmiths, Broad Top and Tioga County, Pa., 
coals are standards; while for coking, Connellsville is recog¬ 
nized as the best. 

Heating Formulas.—A British thermal unit (B. T. U.) is the 
quantity of heat required to raise the temperature of 1 lb. of 
water 1° F. at or near the temperature of maximum density, 
39.1° F. 

A calorie (cal.) is the quantity of heat required to raise the 
temperature of 1 Kg. of water 1° C. at or about 4° C. 

A pound, calorie is the quantity of heat necessary to raise the 
temperature of 1 lb. of water 1° C. 

1 French cal. = 3.968 B. T. U. 

1 B. T. U. =.252 cal. 


1 lb. cal. =f B. T. U. = .4536 cal. 

The heating value of any coal may be calculated from its 
ultimate analysis, with a probable error not exceeding 2%, by 
Dulong’s formula: 


Heating value per pound = 146 C-j-620 



in which C, II, and O are, respectively, the percentages of 
carbon, hydrogen, and oxygen. 


PROXIMATE ANALYSES AND HEATING VALUES OF AMERICAN COALS 


STEAM 


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136 


STEAM 


Heat in pound calorie = 8,080 C+34,462 
or = 8,080 C+34 
Heat in B. T. U. = 14,650 C+62,100 ( H — 




K) 


+2,250 S 

O 
8 


in which C, O, H, and 5 represent the weights of carbon, oxygen, 
hydrogen, and sulphur in 1 lb. of the substance. 


COMPOSITION OF FUELS 


Description 

Carbon 

Per Cent. 

Hydrogen 
Per Cent. 

Oxygen 

Per Cent. 

Nitrogen 
Per Cent. 

Sulphur 

Per Cent. 

Ash 

Per Cent. 

Anthracite 







France. 

90.9 

1.47 

1.53 

1.00 

.80 

4.3 

Wales. 

91.7 

3.78 

1.30 

1.00 

.72 

1.5 

Rhode Island. 

85.0 

3.71 

2.39 

1.00 

.90 

7.0 

Pennsylvania. 

78.6 

2.50 

1.70 

.80 

.40 

14.8 

Semibituminous 







Maryland. 

80.0 

5.00 

2.70 

1.10 

1.20 

8.3 

Wales. 

88.3 

4.70 

.60 

1.40 

1.80 

3.2 

Bituminous 







Pennsylvania. 

75.5 

4.93 

12.35 

1.12 

1.10 

5.0 

Indiana. 

69.7 

5.10 

19.17 

1.23 

1.30 

3.5 

Illinois. 

61.4 

4.87 

35.42 

1.41 

1.20 

5.7 

Virginia. 

57.0 

4.96 

26.44 

1.70 

1.50 

8.4 

Alabama. 

53.2 

4.81 

32.37 

1.62 

1.30 

6.7 

Kentucky. 

49.1 

4.95 

41.13 

1.70 

1.40 

7.2 

Cape Breton. 

67.2 

4.26 

20.16 

1.07 

1.21 

6.1 

Vancouver Island. . 

66.9 

5.32 

8.76 

1.02 

2.20 

15.8 

Lancashire gas coal. 

80.1 

5.50 

8.10 

2.10 

1.50 

2.7 

Boghead cannel. . .. 

63.1 

8.90 

7.00 

.20 

1.00 

19.8 

Lignite 







California brown. . . 

49.7 

3.78 

30.19 

1.00 

1.53 

13.8 

Australian brown... 

73.2 

4.71 

12.35 

1.11 

.63 

8.0 

Petroleum, 







Pennsylvania 







(crude). 

84.9 

13.70 

1.40 




Caucasian (light)... 

86.3 

13.60 

.10 




Caucasian (heavy) . 

86.6 

12.30 

1.10 




Refuse. 

87.1 

11.70 

1.20 




































STEAM 


137 


STEAM BOILERS 

The steam boiler that will be the most suitable for a certain 
mine will depend on the nature of tlie feedwater, the cost of 
fuel, and the amount of steam required. When the acid water 
from the mine is used for feedwater and fuel is cheap, either the 
plain cylindrical or the flue boiler is used, because it is simple 
in construction and can therefore be easily cleaned and cheaply 
replaced when eaten by the mine water. The tubular or loco¬ 
motive type is used where good water can be obtained * except 
in the best-equipped plants, where the water-tube boiler is used. 
Feedwater taken from the mine, or containing acid, should be 
neutralized by lime or soda before being used. In case it con¬ 
tains minerals in solution, a feedwater separator should be 
employed to precipitate the mineral substance before the water 
is allowed to enter the boiler. 

The heating surface of a boiler is the portion of the surface 
exposed to the action of flames and hot gases. This includes, 
in the case of the multi-tubular boiler, the portions of the shell 
below the line of brickwork, the exposed heads of the shell, and 
the interior surface of the tubes. In the case of a water-tube 
boiler, the heating surface comprises the portion of the shell 
below the brickwork, the outer surface of the header^and outer 
surface of tubes. 

Horsepower of Boilers.— The horsepower of a boiler is a 
measure of its capacity for generating steam. Boilermakers 
usually rate the horsepower of their boilers as a certain fraction 
of the heating surface; but this is a very indefinite method, for 
with the same heating surface, different boilers of the same type 
may, under different circumstances, generate different quan¬ 
tities of steam. 

In order to have an accurate standard of boiler power, the 
American Society of Mechanical Engineers has adopted as a 
standard horsepower an evaporation of 30 lb. of water per hour 
from a feedwater temperature of 100° F. into steam at 70 lb. 
gauge pressure, which is considered equivalent to 34.5 units.of 
evaporation; that is, to 34.5 lb. of water evaporated from 
a feedwater temperature of 212° F. into steam at the same 
temperature. 


138 


STEAM 


High-Pressure Steam.—A calculation of the power that 

coal possesses, compared with the useful work which steam 
engines exert, shows that probably in the very best engines 
not one-tenth of the power is converted into useful work, and 
in some very bad engines, probably not one one-hundredth. 
Whatever pressure may be available at the steam boiler, a cer¬ 
tain amount is absorbed in overcoming the resistance of the 
engine and without doing any useful work. Then, again, the 
amount of work that it is possible to get out of a given quantity 
of steam depends on the difference between the temperature 
at the commencement of the stroke and the temperature at the 
end of the stroke. 

There is a limit as to how low the temperature can be at the 
end, therefore, as the commencing temperature is raised the 
available difference is increased. The advantages of high- 
pressure steam may be shown by taking a fixed temperature 
in the condenser of say, 100° F., then initial temperatures 
when the steam enters the cylinder, the temperature of varying 
amounts, and the theoretic efficiency of that steam can be deter¬ 
mined. At atmospheric pressure, there is an efficiency of 
16.6%. 


EFFICIENCY OF STEAM AT VARIOUS PRESSURES 


Steam 

Pressure 

Pounds 

Efficiency 

Per Cent. 

Steam 

Pressure 

Pounds 

Efficiency 

Per Cent. 

10 

20.0 

100 

29.8 

20 

22.1 

125 

31.1 

30 

23.7 

150 

32.2 

40 

25.0 

200 

33.9 

50 

26.1 

250 

35.3 

60 

27.0 

300 

36.5 

80 

28.6 




In practice, only a certain proportion of the theoretic power 
of steam can be obtained, and that proportion varies with the 
pressure of the steam. The advantages of high-pressure steam 
are not yet sufficiently appreciated. It is not merely the 
difference between 60 lb. and 120 lb. Suppose we use steam 











STEAM 


139 


at 60 lb.; probably we shall get 50 lb. a t engine, and resist¬ 
ances of engine will absorb 10 lb., leaving 40 lb. Now, sup¬ 
pose we use 120 lb., we can get at engine 110 lb., and if 
resistances of engine absorb 10 lb., we shall have 100 lb. as 
against 40 lb. 

By expansion of steam is meant that at a certain point of the 
stroke, the steam supply from the boiler to the cylinder is shut 
off and the steam already within the cylinder performs the 
remainder of the stroke unaided. 

Incrustation.—Nearly all waters contain foreign substances 
in a greater or less degree, and though this may be a small 
amount in each gallon, it becomes of importance where large 
quantities are evaporated. For instance, a 100-H. P. boiler 
evaporates 30,000 lb. of water in 10 hr. or 390 T. per mo.; in 
comparatively pure water there should be 88 lb. of solid matter 
in that quantity, and in many kinds of spring water as much as 
2,000 lb. 

The nature and hardness of the scale formed will depend 
on the kind of substances held in solution and suspension. 
Analyses of incrustations show that carbonate and sulphate of 
lime form the larger part of all ordinary scale, that from car¬ 
bonate being soft and granular, and that from sulphate, hard 
and crystalline. Organic substances in connection with carbon¬ 
ate of lime will also make a hard and troublesome scale. The 
causes of incrustation are: 

1. Deposition of suspended matter. 

2. Deposition of salts from concentration. 

3. Deposition of carbonates of lime and magnesia, by boiling 
off carbonic acid, which holds them in solution. 

4. Deposition of sulphates of lime, because sulphate of lime 
is soluble in cold water, less soluble in hot water, insoluble 
above 270° F. 

5. Deposit of magnesia, because magnesium salts decompose 
at high temperatures. 

6. Deposition of lime soap, iron soap, etc. formed by sapon¬ 
ification of grease. 

Incrustation may be prevented by the following methods: 

1. Filtration. 

2. Blowing off. 


140 


STEAM 


3. Use of internal collecting apparatus, or devices, for 
directing the circulation. 

4. Heating feedwater. 

5. Chemical or other treatment of water in boiler. 

6. Introduction of zinc in boiler. 

7. Chemical treatment of water outside of boiler. 


INCRUSTATION REMEDIES 


Troublesome Substances 

Trouble 

Remedy or Palliation 

Sediment, mud, clay, etc. 

Incrustation 

Filtration; blowing off 

Readily soluble salts. . . . 

Incrustation 

Blowing off 

Bicarbonate of lime, 


magnesia, and iron.... 

Incrustation 

Heating feed; addition 
of caustic soda, lime, 
or magnesia, etc. 

Sulphate of lime. 

Incrustation 

Addition of carbonate 
of soda, barium chlo¬ 
ride, etc. 

Chloride and sulphate of 


magnesium.... . 

Corrosion 

Addition of carbonate 
soda, etc. 

Carbonate of soda in 


large amounts. 

Priming 

Addition of barium 
chloride, etc. 

Acid in mine water. 

Corrosion 

Alkali 

Dissolved carbonic acid 



and oxygen. 

Corrosion 

Heating feed; addi¬ 
tion of caustic soda, 
slaked lime, etc. 

Grease, from condensed 


water. 

Corrosion 

Slaked lime and filter¬ 
ing; substitute min¬ 
eral oil 

Organic matter, sewage. 

Priming 

Precipitate with alum 
or ferric chloride, 
and filter 

Organic matter. 

» 

Corrosion 

Precipitate with alum 
or ferric chloride, 
and filter 


Prevention of Incrustation.—The incrustation of boilers may 
be prevented by adding various substances to the feedwater. 
Oak, hemlock, sumac, catechu, logwood, and other barks and 

















STEAM 


141 


woods, are effective in waters containing carbonates of lime 
or magnesia, by reason of their tannic acid, but are injurious 
to the iron and not to be recommended. 

Molasses, cane juice, vinegar, fruits, distillery slops, etc., have 
been used, but the acetic acid that they contain is even more 
injurious to the iron than tannic acid, while the organic matter 
forms a scale with sulphate of lime when it is present. 

Milk of lime and metallic zinc have been used with success in 
waters charged with bicarbonate of lime, reducing the bicar¬ 
bonate to the insoluble carbonate. 

Barium chloride and milk of lime are said to be used, with 
good effect at Krupp’s works, in Prussia, for waters impregnated 
with gypsum. 

Soda ash and other alkalies are very useful in waters con¬ 
taining sulphate of lime, by converting it into a carbonate, and 
so forming a soft scale that is easily cleaned. But when used 
in excess they cause foaming, particularly where there is oil 
coming from the engine, with which they form soap. All soapy 
substances are objectionable for the same reason. 

Petroleum has been much used of late years; it acts best in 
waters in which sulphate of lime predominates. Sulphate of 
lime is the injurious substance in nearly all mine waters, and 
petroleum, when properly prepared, is a good preventive of 
scale and pitting. Crude petroleum should not be used, as 
it sometimes helps to form a very injurious scale. Refined 
petroleum, on the other hand, is useless, as it vaporizes at 
a temperature below that of boiling water. Therefore, only 
such preparations should be used as will not vaporize below 
500° F. 

Tannate of soda is a good preparation for general use, but in 
waters containing much sulphate, it should be supplemented by 
a portion of carbonate of soda or soda ash. 

A decoction from the leaves of the eucalyptus is found to work 
well in some waters in California. 

For muddy water, particularly if it contain salts of lime, no 
preventive of incrustation will prevail except filtration, and in 
almost every instance the use of a filter, either alone or in con¬ 
nection with some means of precipitating the solid matter from 
solution, will be found very desirable. 


142 


STEAM 


In all cases where impure or hard waters are used, frequent 
blowing from the mud-drum is necessary to carry off the accu¬ 
mulated matter, which if allowed to remain would form scale. 

When boilers are coated with a hard scale, diffcult to remove, 
the addition of i lb. caustic soda per horsepower, and steaming 
for some hours, according to the thickness of the scale, will 
greatly facilitate the cleaning, rendering the scale soft and loose. 
This should be done, if possible, when the boilers are not other¬ 
wise in use. 


STEAM ENGINES 

Requirements of a Good Steam Engine.—A good steam 
engine should be as direct acting as possible; that is, the con¬ 
necting parts between the piston and the crank-shaft should 
be few in number, as each part wastes some power. Formerly, 
beam engines were in general use and were suitable for pump¬ 
ing when the pump was at one end of the beam and the piston 
at the other. Few of modern colliery engines, however, are 
thus equipped. The moving parts of an engine should be 
strong, to resist strains, and light, so as to offer no undue resist¬ 
ance to motion; parts moving upon each other should be well 
finished, to reduce resistances to a minimum; the steam should 
get into the cylinder easily at the proper time, and the exhaust 
should leave the cylinder as exactly and as easily. The steam 
pipes supplying steam should have an area one-tenth the com¬ 
bined areas of the cylinders they supply, and exhaust pipes 
should be somewhat larger. The cylinder, steam pipes and 
boiler should be well protected. The engine should be capable 
of being started and stopped and reversed easily and quickly. 

Rule .—To find the indicated horsepower developed by an engine, 
multiply the mean effective pressure per square inch, the area oj 
the piston, the length of stroke, and the number of strokes per 
minute; this gives the work per minute in foot-pounds. Divide 
the product by 33,000. 

Let I. H. P. = indicated horsepower of engine; 

P=M. E. P., in pounds per square inch; 

A =area of piston, in square inches; 

L = length of stroke,' in feet; 

N — number of strokes per minute. 



STEAM 


143 


Then, 


I. H. P. = 


PLAN 


33,000 

The number of strokes per minute is twice the number of 
revolutions per minute. For example, if an engine runs at a 
speed of 210 rev. per min., it makes 420 strokes per minute. A 
few types of engines, however, are single acting; that is, the 
steam acts on only one side of the piston, then the number of 
strokes per minute equals the number of revolutions per minute. 

Approximate Determination of M. E. P.—To approximately 
determine the mean effective pressure, M. E. P., of an engine, 
when the point of apparent cut-off is known and the boiler 
pressure, or the pressure per square inch in the boiler from 
which the supply of steam is obtained, is given: 

Rule. —To find the M. E. P. of good, simple, non-condensing 
engines, add 14-7 to the gauge pressure, and multiply the result by 
the number opposite the fraction indicating the point of cut-off 
in the accompanying table. Subtract 17 from the product, and 
multiply by .9. 

Let p = gauge pressure; 

k = a constant; 

M. E. P. = mean effective pressure. 

Then, M. E. P. = .9X[&(/> + 14.7) — 17] 


TABLE OF CONSTANTS 


Cut-Off 

Constant 

Cut-Off 

Constant 

Cut-Off 

Constant 

1 

6 

.566 

3 

8 

.771 

2 

3 

.917 


.603 

I 

.789 


.926 

1 

4 

.659 

1 

3 

.847 

a 

4 

.937 

A 

.708 

i 

.895 


.944 

1 

3 

.743 

5 

8 

.904 

7 

8 

.951 


If the engine is a simple condensing one, subtract the pressure 
in the condenser instead of 17. The fraction indicating the 
point of cut-off is obtained by dividing the distance that the 
piston has traveled when the steam is cut off by the whole 
length of the stroke. For a f cut-off, and 92 lb. gauge pres¬ 
sure in the boiler, the M. E. P. is, by the formula just given, 
.9X[.917X(92-H4.7)-17] = 72.7 lb. per sq. in. 


















144 


COMPRESSED AIR 


COMPRESSED AIR 

An air compressor consists essentially of a cylinder in which 
atmospheric air is compressed by a piston, the driving power 
being steam or water. Steam-driven compressors in ordinary 
use may be classed as follows: 

(1) Straight-line type, in which a single horizontal air 
cylinder is set tandem with its steam cylinder, and provided 
with two flywheels; this pattern is generally adapted for com¬ 
pressors of small size. 

(2) Duplex type, in which there are two steam cylinders, 
each driving an air cylinder, and coupled at 90° to a crank¬ 
shaft carrying a flywheel. 

(3.) Horizontal, cross-compound engines, each steam cyl¬ 
inder set tandem with an air cylinder. 

(4.) Vertical, simple or compound engines, with the air 
cylinders set above the steam cylinders. 

(5.) Compound or stage compressors, in which the air cyl¬ 
inders themselves are compounded. The compression is car¬ 
ried to a certain point in one cylinder and successively raised 
and finally completed to the desired pressure in the others. 
They may be either of the straight-line or duplex form, with 
simple or compound steam cylinders. 

The first three and the last classes are those commonly used 
for mine service. The principle of compound, or two-stage, 
air compression is recognized as applicable for even the moder¬ 
ate pressures required in mining, and the compressors of 
class 5 are frequently employed. 

Transmission of Air in Pipes.—The actual discharge capacity 
of piping is not proportional to the cross-sectional area alone, 
that is, to the square of the diameter. Although the periphery 
is directly proportional to the diameter, the interior surface 
resistance is much greater in a small pipe than in a large 
one, because, as the pipe becomes smaller, the ratio of 
perimeter to area increases. Among the formulas in com¬ 
mon use, perhaps the most satisfactory is that of D’Arcy. 
As adopted for compressed-air transmission, it takes the 
form: 



COMPRESSED AIR 


145 


_ . U^ipi — pi) 

D = c\ -, 

\ wil 

in which D = volume of compressed air, in cubic feet per min¬ 
ute, discharged at final pressure; 
c = coefficient varying with diameter of pipe, as 
determined by experiment; 
d = diameter of pipe, in inches (the actual diameters 
of 1| in. and 1| in. pipe are 1.38 in. and 1.61 
in., respectively; the nominal diameters of all 
other sizes may be taken for calculations); 
l — length of pipe, in feet; 

pi= initial gauge pressure, in pounds per square inch; 
p 2 ~final gauge pressure, in pounds per square inch; 
wi = density of air, or its weight, in pounds per cubic 
foot, at initial pressure pi. 

The values of the coefficients c for piping up to 12 in. in 
diameter are given in the accompanying table. Some appar¬ 
ent discrepancies exist for sizes larger than 9 in. but they cause 
no very material differences in the result. 


PIPING COEFFICIENTS 


Size of 
Pipe 

Inches 

Coeffi¬ 
cient c 

Size of 
Pipe 

Inches 

Coeffi¬ 
cient c 

Size of 
Pipe 

Inches 

Coeffi¬ 
cient c 

1 

45.3 

5 

59.0 

9 

61.0 

2 

52.6 

6 

59.8 

10 

61.2 

3 

56.5 

7 

60.3 

11 

61.8 

4 

58.0 

8 

60.7 

12 

62.0 


Loss of Pressure in Transmission.—In the accompanying 
table is given the loss of pressure in the transmission of com¬ 
pressed air, calculated for pipes 1,000 ft. long. For other 
lengths the loss varies directly as the length. The resistance 
is not varied by the pressure, only so far as changes in pres¬ 
sure vary the velocity. It increases about as the square of 
the velocity, and directly as the length. Elbows, short turns, 
and leaks in pipes tend to reduce the pressure in addition to 
the losses given in the table. 

















LOSS OF PRESSURE BY FLOW OF AIR IN PIPES 


146 


COMPRESSED AIR 


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ELECTRICITY 


147 


ELECTRICITY 


ELECTRIC GENERATORS AND MOTORS 

An electric generator is a machine for converting mechanical 
energy into electrical energy. An electric motor is a machine 
for converting electrical energy into mechanical energy. Either 
a generator or a motor may be called a dynamo; but this word 
is commonly used to denote generators only. Generators and 
motors may be divided into two general classes: those used 
with direct current and those used with alternating current. 

Direct-Current Generators. —Direct-current generators are 
those that furnish a current always in the same direction. 
They have three essential parts: the field magnet, often called 
the field, the armature, and the commutator; the field is sta¬ 
tionary but the armature revolves. The coils of wire on the 
field are called the field coils, and when properly connected 
make up the field winding. The individual coils on the arma¬ 
ture are called the armature coils, and when properly connected 
make up the armature winding. The commutator consists 
of copper bars arranged in the form of a cylinder and revolves 
with the armature. Stationary brushes, usually of carbon, 
collect the current from the commutator; these brushes are 
held in brush holders. 

Direct-Current Motors. —Direct-current motors are in gen¬ 
eral almost identical, so far as construction goes, with direct- 
current generators. Motors are often required to operate 
under very trying conditions, as for example, in mine haulage 
or pumping plants or on the ordinary street car. For this 
reason, their mechanical construction often differs from that 
of the generator so that the working parts will be enclosed as 
completely as possible, and thus protect them from dirt and 
injury. Practically all of the motors in use are operated 
from constant-pressure mains; i. e., the pressure at the terminals 
of the motor is practically constant, no matter what load it 
may be carrying. The various principal parts of a direct- 
current motor and a generator are similarly named. 



148 


ELECTRICITY 


None but very small motors or motors of special design 
should be started without some form of resistance in series 
with it to cut down the electric pressure of the line until the 
motor is up to speed. Such a resistance is called a starting 
rheostat, or starting box. 

Motor Troubles.—If a motor fails to start when the con¬ 
trolling switch is closed, the trouble may be due to any one of 
several causes. There may be an open circuit, a short circuit, 
a wrong connection, the power may be off the line, or the 
trouble may be purely mechanical. If there is no power on 
the line, the lamps on the same line will be out. If the power 
is on the line, but there is no flash at the starting box when 
the handle is moved on and then off, the trouble is due to an 
open circuit. Open circuits that may be located by inspec¬ 
tion are: defective switches, broken wires, loose or open con¬ 
nections, something under one of the brushes, brush stuck 
in the holder, brush dropped out of holder, or a blown fuse. 
With the exception of crosses in the wiring, short circuits are 
usually in the motor itself and require the attention of a 
skilled electrician. 

Mechanical troubles that may interfere with the starting of 
a motor are sometimes overlooked. The more common are: 
too much load, bearings worn until the armature rubs the field 
magnet, sprung armature shaft, hot-box, tight belt or some¬ 
thing in the gearing, lack of end play in the armature. 

Excessive flashing at the brushes of a motor is called spark¬ 
ing. When a motor sparks badly the attention of an electri¬ 
cian should be called to it. The following are some of the com¬ 
mon causes of sparking: Too much load, brushes improperly 
set, commutator rough or eccentric, dirty brushes or commu¬ 
tator, loose brushes, sprung armature shaft, low bearings, worn 
commutator, vibration, belt slipping. There are other causes, 
but they cannot be located readily except by an electrician. 

Heating of a motor is usually due to overload. If a person 
cannot hold his hand firmly on any part of a motor, the machine 
is running too hot. Most modem motors will run continu¬ 
ously under an overload of 25% without serious heating and 
will run for 2 hr. under 50% overload without damage due to 
heating. These limits should not be exceeded. 


ELECTRICITY 


149 


Alternating-Current Generators.—An alternating-current 
generator, commonly called an alternator, is one that estab¬ 
lishes a current which periodically reverses its direction. 
Alternators are now largely used for both lighting and power 
transmission, especially when the transmission is over long 
distances. Alternating current is specially suitable for long¬ 
distance work because it may be readily transformed from one 
pressure to another. In order to keep down the amount of 
copper in the line, a high line pressure must be used. Pressures 
much over 500 or 600 volts cannot be readily generated with 
direct-current machines, owing to the troubles likely to arise 
due to sparking at the commutator. An alternator requires 
no commutator; usually the armature is made stationary and 
the field revolving. Alternators are now built that generate 
as high as 8,000 or 10,000 volts directly. If a still higher pres¬ 
sure is required on the line, it can be easily obtained by the 
use of transformers. 

Alternators may be divided into two classes: Single-phase 
and polyphase alternators. Single-phase alternators are so 
called because they set up a single alternating current. Poly¬ 
phase alternators are so called because they deliver two or 
more alternating currents that differ in phase; that is, currents 
that do not reach their maximum nor their zero values of the 
same direction at the same instant. 

Polyphase alternators are well adapted for power and light¬ 
ing purposes in mines, especially for the operation of pumping 
and hoisting machinery, because the motors operated by them 
are simple in construction and not liable to get out of order. 

Alternating-Current Motors.—Alternating-current motors 
may be divided into two classes: synchronous and induction 
motors. Synchronous motors are almost identical, so far as 
construction goes, with the corresponding alternator. Induc¬ 
tion motors are so called because the current is induced in the 
armature instead of being led into it from some outside source. 
The stationary part of an induction motor is usually called 
the stator and the revolving part, the rotor. 

Induction motors possess many advantages for mine work. 
One is the absence of the commutator or any kind of sliding 
contacts whatever. Such motors can therefore operate with 


150 


ELECTRICITY 


absolutely no sparking—a desirable feature for mine work. 
The motors are also very simple in construction, and are not 
liable to get out of order, but, like direct-current motors, 
they should not be overloaded 

Transformers.—Transformers are used to change an alter¬ 
nating current from a higher to a lower pressure, or vice versa, 
with a corresponding change in current. Transformers used 
for raising the voltage are known as step-up transformers; 
those used for lowering the pressure are known as step-down 
transformers. The transformer consists of a laminated iron 
core upon which are wound two coils of wire that have no 
connection with each other. One of these coils, called the 
primary, is connected to the mains; the other coil, called the 
secondary, is connected to the circuit to which current is deliv¬ 
ered. The core and coils are contained in an iron case usually 
filled with oil. 


ELECTRIC CIRCUITS 

The path through which a current flows is generally spoken 
of as an electric circuit. This path may be made up of a num¬ 
ber of parts. For example, the line wires may constitute 
part of the circuit, and the remainder may be composed of 
lamps, motors, resistances, etc. In practice, the two kinds 
of circuits most commonly met with are those in which the 
different parts of the circuit are connected in series and those 
in which the different parts of the circuit are connected in 
multiple or parallel. 

Series Circuits.—In the series circuits any two adjacent 
parts are connected in tandem, so that the current passing 
through one part also passes through the other parts. Fig. 1 
(a) represents such a circuit in which the current passes from 
the generator D at the+side through the arc lamps a, through 
the incandescent lamps l, through the motor m and the resist¬ 
ance r, back to the generator D. The most common use of 
this system is in connection with arc lamps. 

The objections to this system of distribution for general 
work are that the breaking of the circuit at any point cuts 
off the current from all parts of the circuit; also, the pressure 



ELECTRICITY 


151 


generated by the dynamo has to be very high if many pieces 
of apparatus are connected in series. In such a system, the 
dynamo is provided with an automatic regulator that increases 
or decreases the voltage of the machine, so that the current 
in the circuit is kept constant, no matter how many lamps 
or other devices are in operation. For this reason, such 
circuits are often spoken of as constant-current circuits. 

A series circuit should never be opened at any point unless 
it is known that there is no current in the line. If it is desired 



(*>) 

Tro//ey W/re 



B 

(d) 

Fig. 1 


to disconnect an arc lamp, for instance, from a series circuit, 
one end of a short wire, called a jumper, should be connected 
to the line wire on each side of the lamp, so that the cur¬ 
rent may pass through the jumper. Then the lamp may be 
disconnected. 

Parallel Circuits.—In parallel circuits, the different pieces of 
apparatus are connected side by side, or in parallel, across 
the main wires, as shown in Fig. 1 ( b ). In this case, the 
generator D supplies current through the mains to the arc 




















152 


ELECTRICITY 


lamps a, incandescent lamps l, and motor m. This system is 
more widely used as the breaking of the circuit through any 
one piece of apparatus will not stop the current through the 
other parts. Incandescent lamps are connected in this way 
almost entirely. Street cars and mining locomotives are oper¬ 
ated in the same way, the trolley wire constituting one main 
and the track the other, as shown in Fig. 1 (c). By adopting 
this system, any car can move independently of the others, 
and the current may be turned off and on at will. In all these 
systems of parallel distribution, the pressure of the generator 
is maintained constant no matter what current the generator 
may be delivering. In mine-haulage plants, the pressure is 
usually 250 or 500 volts, the former being generally preferred 
as being less dangerous. Lamps may also be connected in 
series-multiple, as shown in Fig. 1 (e). Here the two 125-volt 
lamps l are connected in series across the 250-volt circuit. Such 
an arrangement is frequently used in mines when lamps are 
operated from the haulage circuit. Parallel circuits are called 
constant-potential circuits, to distinguish them from the constant- 
current circuit mentioned previously. 

Shunt.—When one circuit B, Fig. 1 ( d ), is connected across 
another A, so as to form, as it were, a by-pass, or side-track, 



for the current, such a circuit is called a shunt, or it is said to 
be in shunt with the other circuit. 

Distribution Systems.—Electric circuits may also be clas¬ 
sified as direct-current circuits or alternating-current circuits, 
depending on the kind of current carried. The system of 
conductors, or wires, leading from a power station is called a 
distribution system: it is a direct-current system if direct current 
is used, an alternating-current system if alternating current is 
used. 











ELECTRICITY 


15S 


Direct-current is usually distributed by either the two-wire 
system, shown in Fig. 2, or the three-wire system, shown in Fig. 3. 

Alternating current may be distributed by the single-phase 
system, the two-phase system, or the three-phase system. The 
single-phase system usually employs two wires, as in Fig. 2; 
the two-phase system, four wires; and the three-phase system, 
three wires. The three-phase system is similar to the direct- 
current three-wire system in the number of wires only. The 
single-phase system is rarely used for mines. 



Fig. 3 



Protection of Circuits.—It is necessary to protect electrical 
apparatus from the danger of burn-outs due to heavier currents 
than those for which they are designed. This is accomplished 
by means of cut-outs, which automatically open a circuit when 
the current exceeds a certain value. A cut-out may be either 
a fuse or a circuit-breaker. 

Electric-Haulage Circuits.—In electric-haulage circuits, the 
rails take the place of one of the conductors, so that, in cal¬ 
culating the size of feeders required, only the overhead con¬ 
ductors are taken into account. It is difficult to assign any 
definite value to the resistance of the track circuit, as this 
resistance depends very largely on the quality of the rail 
bonding at the joints. If this bonding is well done, the resist¬ 
ance of the return circuit should be very low, because the 
cross-section of the rails is comparatively large. The follow¬ 
ing example will serve to illustrate how calculations for haul¬ 
age circuits are made. 
























154 


ELECTRICITY 


Example. —In Fig. 4 a b represents a section of track 4,000 ft. 
long. From the dynamo c to the beginning of the section, the 
distance is 1,200 ft. The trolley wire is No. 00 B. &. S., and 
is fed from the feeder at regular intervals. Two mining loco¬ 
motives are operated, each of which takes an average current 
of 75 amp. The total allowable drop to the end of the line is 
to be 5% of the terminal voltage, which is 500 volts. Calculate 
the size of feeder required, assuming that the constant 14, in 
the formula, takes account of the resistance of the return circuit. 

Solution. —As the locomotives are moving from place to 
place, the center of distribution for the load may be taken 
at the center of the 4,000 ft. The distance L will then be 
l,200-}-2,000 = 3,200 ft. The total current will be 150 amp.; 

14X3,200X150X100 

hence,-= 268,800 cir. mis. 

500X500 



Fig. 4 


This would require either a stranded cable or the use of two 
No. 00 wires in parallel from c to a. From a to b, the No. 00 
trolley wire is in parallel with the feeder; hence, the section 
of feeder a b may be a single No. 00 wire. 

In many cases, the drop is allowed to run as high as 10%, 
because the loads are usually heavier, and the distances longer, 
than in the example just given. 


ELECTRIC APPARATUS IN FIREDAMP 

If a fuse blows or a motor sparks in firedamp, the gas will be 
ignited just as surely as though it came in contact with a naked 
lamp. In any part of a mine where firedamp is apprehended, 
a safety lamp should be provided for use with each electric 
machine when working, and should the safety lamp give any 
indication of gas, the person in charge should immediately 























ELECTRICITY 


155 


stop the machine, cut off the current at the nearest switch, 
and report the matter to the proper authority. This applies 
especially to such apparatus as electrically driven coal cutters. 
The operator of electric machinery should never leave the 
machine running; that is, with the current on. 


SIMPLE ELECTRICAL CALCULATIONS 

PRACTICAL UNITS 

In electrical work it is necessary to have units in terms of 
which to express the different quantities entering into calcula¬ 
tions. The four most important of these are used to express 
current, electrical pressure, or electromotive force, resistance, 
and power. 

Current. —The current in a wire may be indicated in sev¬ 
eral ways. If a compass needle is held under or over a wire, 
it will be deflected and will tend to stand at right angles to 
the wire. The stronger the current, the greater will be the 
deflection of the needle. The unit used to express current 
is called the ampere. The expression of current through a wire 
as so many amperes is analogous to the expression of the flow 
of water through a pipe as so many gallons per second. 

Electromotive Force. —In order that a current may pass 
through a wire, there must be an electrical pressure of some 
kind to cause the flow, just as in hydraulics there must always 
be a head or pressure before water can be made to flow through 
a pipe. However, there may be a pressure or head without 
there being any flow of water, because the opening in the 
pipe might be closed, though as soon as the valve closing the 
pipe is opened, the current will start. In the same way, 
an electrical pressure or electromotive force (usually written 
E. M. F.) may exist in a circuit, but no current can pass until 
the circuit is closed or until the wire is connected so that there 
will be a path for the current. The practical unit of electro¬ 
motive force is the volt. It is the unit of electrical pressure, 
and fulfills somewhat the same purpose as pounds per square 
inch in hydraulic and steam engineering. 



MECHANICAL EQUIVALENTS OF ELECTRICAL UNITS 


156 


ELECTRICITY 
























ELECTRICITY 


157 


Resistance.—All conductors offer more or less resistance 
to a current of electricity, just as water encounters friction 
in passing through a pipe. The amount of this resistance 
depends on the length of the wire, the diameter of the wire, 
and the material of which the wire is composed. The resist¬ 
ance of all metals also increases with the temperature. The 
practical unit of resistance is the ohm. A conductor has a resist¬ 
ance of 1 ohm when the pressure required to set up 1 ampere 
through it is 1 volt. In other words, the drop, or fall, in pressure 
through a resistance of 1 ohm, when a current of 1 amp. is pas¬ 
sing, is 1 volt. 1,000 ft. of copper wire .1 in. in diameter has 
a resistance of nearly 1 ohm at ordinary temperatures. 

Power.—-The electrical unit of power' is the watt. It is 
equal to the power developed by a current of 1 amp. under a 
pressure of 1 volt. The watt is equal to H. P. The watt 
is too small a unit for convenient use in many cases, so that 
the kilowatt, equal to 1,000 watts, is frequently used. The 
word kilowatt is sometimes abbreviated to kw. or K. W. 

Work.—The electrical unit of work is the watt-hour. This 
is the total work done at the rate of 1 watt for 1 hr. A kilo¬ 
watt hour is 1,000 watt-hours. It is equivalent to W. or 
about li H. P.-hr. 0HM , S LAW 


Ohm's law may be briefly stated as follows: 

Law.— The strength of the current in any circuit is directly 
proportional to the E. M. F. in the circuit, and inversely pro¬ 
portional to the resistance of the circuit. 

This means that if the resistance of a circuit is fixed, and 
the E. M. F. varied, the current will be doubled if the E. M. F. 
is doubled. Also, if the E. M. F. is fixed, and the resistance 
doubled, the current will be halved. 

Let E = E. M. F., in volts; 

R = resistance, in ohms; 

7 = current, in amperes. 

F E 

Then, / = —, or R = ~, or E = IR 

R I 

The last two forms are useful in many cases where the usual 
E 

form I = — is not directly applicable. 

R 


158 


ELECTRICITY 


D )E-110 Volts 


Example 1.—In the accompanying figure, a dynamo D 
that generates 110 volts, is connected to a coil or wire C, 

which has a resistance of 20 ohms; 
what current will pass, supposing the 
resistance of the rest of the circuit to be 
negligible? 

Solution. —Here, E = 110 volts; R 
110 

= 20 ohms; hence, / =-= 5.5 amp. 

20 

Example 2. —If the resistance of the coil C is 6 ohms, what 
E. M. P. must the dynamo generate in order to set up a current 
of 15 amp. through it? 

Solution. —The third form of the law given is more con- 
venient in this case. £ _ 15x 6 , 90 volt8 


£>1 

O.SS 

p© 

CP® 

O II 

-ST6$ 


In case the current and E. M. F. are known, the resistance 
of the circuit may be calculated by using the second form of 
the law given above. For example, if the current in the 
example just given were 8 amp. and the E. M. F. of the dynamo 
110 volts, the resistance of the circuit must be 

110 

R =-= 13.75 ohms 

8 


POWER CALCULATIONS 

Power in Direct-Current Circuits. —The power in any direct- 
current circuit may be found by multiplying the current by 
the pressure required to maintain the current in the circuit. 

Let W = power, in watts; 

H. P. = horsepower. 

Then, watts = voltsX amperes, or 

W = EI 
W=PR 
IV 

H. P. =- 

746 

Power in Alternating-Current Circuits. —Owing to the nature 
of alternating currents the formula W = E I cannot always 
be used for alternating-current calculations. It may be used 
for single-phase circuits to which nothing but incandescent 





ELECTRICITY 


159 


lamps are connected. On single-phase circuits operating 
motors only the formula becomes approximately W= .8 El, 
and on single-phase circuits containing both lamps and motors 
the formula W = .85 E I may be used. For two-phase circuits 
the foregoing expressions will give the power for one phase, 
so that to obtain the total power for both phases multiply 
by 2. On three-phase circuits, the expressions for power are 
W = 1.732 E I, W = 1.386 E I, and W = 1.472 E /, respectively, 
for the three conditions mentioned. 

WIRE DATA 

Estimation of Cross-Section of Wires.—The diameter of 
round wires is usually given in the tables in decimals of an 
inch, and the area of cross-section is given in terms of a unit 
called circular mil. This is done simply for convenience in 
calculation, as it makes calculations of the cross-section 
much simpler than if the square inch were used as the unit 
area. A mil is ^ns in., or .001 in. A circular mil is the area 
of a circle, the diameter of which is ti&tt in., or 1 mil, or 
.0000007854 sq. in. 

If the diameter of the conductor were 1 in., its area would 
be .7854 sq. in., and the number of circular mils in its area 
would be .7854-f-.0000007854 = 1,000,000; but 1 in. = 1,000 
mils, and (1,000) 2 = 1,000,000; hence, the following is true: 
C. M. = d 2 - or the area of cross-section of a wire, in circular 
mils, is equal to the square of its diameter expressed in mils. 

Example. —If a wire has a diameter of .101 in., what is its 
area in circular mils? 

Solution. — .101 in. = 101 mils; hence, C. M. = (101) 2 

= 10 , 201 . 

Estimation of Resistance.—The resistance of any conductor 
is directly proportional to its length, and inversely proportional 
to its area of cross-section, or 

L 

R = K-, 

A 

in which R = resistance; 

L = length; 

A =area of cross-section; 

K = constant. 


160 


ELECTRICITY 


If L is expressed in feet and A is expressed in circular mils, 
the constant K must be the resistance of 1 ft. of the wire in 
question of 1 cir. mil cross-section. The resistance of 1 mil-ft. of 
copper wire at 75° F. is about 10.8 ohms. Hence, for copper 

10.8L 

wire, R — -; but A=d 2 when d is the diameter in mils; 

A 


hence, 


R = 


10.8 L 


This formula is easily remembered, and is very convenient 
for estimating the resistance of any length of wire of given 
diameter when a wire table is not at hand, or when the diameter 
of the given wire does not correspond to anything given in 
the table. 

Example. —Find the resistance of 1 mi. of copper wire 
.20 in. in diameter. 

Solution. — 1 mi. = 5,280 ft., .20 in. = 200 mils. Area of 

cross-section = (200) 2 = 40,000 cir. mils. Hence, 


R = 


10.8X5,280 

-= 1.42 ohms 

40,000 


Wire Gauge.—The gauge most generally used in America 
to designate the different sizes of copper wire is the American, 
or Brown & Sharpe (B. &. S.). The sizes given by this gauge 
range from No. 0000, the largest, .460 in. diameter, to No. 40, 
the finest, .003 in. diameter. Wire drawn to the sizes given 
by this gauge is always more readily obtained than sizes 
according to other gauges; hence, in selecting line wire for 
any purpose it is always desirable, if possible, to give the size 
required as a wire of the B. &. S. gauge. A wire can usually 
be selected from this gauge, which will be suited to ordinary 
wiring requirements in and about mines. 

In the accompanying table, the common sizes of wire are 
listed. Wires smaller than No. 14 are not given, because 
No. 14 is the smallest size ordinarily used for light wiring. 
Trolley wires are usually made of either No. 00 or No. 0000. 
Conductors larger than No. 0000 are stranded and have no 
gauge number. Stranded conductors, called jzables, are speci¬ 
fied by their area in circular mils. 





ELECTRICITY 


161 


The table gives the dimensions, weight, and resistance of 
pure copper wire. The weights given are for bare wire. The 
first column gives the B. &. S. gauge number, the second the 
diameter in mils; the diameter in inches is the number given 


PROPERTIES OF COPPER WIRE 


B. & S. Gauge 

Number 

Diameter 

Mils. 

e* 

75 

^ U 

d| 

Weight per 1,000 Ft. 
Pounds 

Weight per Mile 

Pounds 

Resistance per 

1,000 Ft. at 68° F. 
International Ohms 

Current 

Capacity 

N ational 
Board 
Fire 
Under¬ 
writers 
(Amperes) 

W eather- 

Proof 

Rubber- 

Covered 

0000 

460.0 

211,600.0 

640.50 

3,381.4 

.0489 

312 

210 

000 

409.6 

167,805.0 

508.00 

2,682.2 

.0617 

262 

177 

00 

364.8 

133,079.4 

402.80 

2,126.8 

.0778 

220 

150 

0 

324.9 

105,534.5 

319.50 

1,686.9 

.0981 

185 

127 

1 

289.3 

83,694.2 

253.30 

1,337.2 

.1237 

156 

107 

2 

257.6 

66,373.0 

200.90 

1,060.6 

.1560 

131 

90 

3 

229.4 

52,634.0 

159.30 

841.1 

.1967 

110 

76 

4 

204.3 

41,742.0 

126.40 

667.4 

.2480 

92 

65 

5 

181.9 

33,102.0 

100.20 

529.1 

.3128 

77 

54 

6 

162.0 

26,250.5 

79.46 

419.5 

.3944 

65 

46 

7 

144.3 

20,816.0 

63.02 

332.7 

.4973 



8 

128.5 

16,509.0 

49.98 

263.9 

.6271 

46 

33 

9 

114.4 

13,094.0 

39.63 

209.2 

.7908 



10 

101.8 

10,381.0 

31.43 

165.9 

.9972 

32 

24 

11 

90.7 

8,234.0 

24.93 

131.6 

1.2570 



12 

80.8 

6,529.9 

19.77 

104.4 

1,5860 

23 

17 

13 

71.9 

5,178.4 

15.68 

82.8 

1.9990 



14 

64.1 

4,106.8 

12.43 

76.2 

2.5210 

16 

12 


in this column, divided by 1,000. The third column gives the 
area in circular mils, the numbers in this column being equal 
to the squares of those in the second column. The safe car¬ 
rying capacity is also given. 





















162 


ELECTRICITY 


Aluminum Wire.—The resistance of aluminum wire is prac¬ 
tically If times that of copper wire. For insulated (covered) 
aluminum wire, the safe-carrying capacity may be taken as 
84% that of copper wire with the same kind of insulation. 
To estimate the resistance of aluminum wire use the formula 
for copper and multiply the result by If. To obtain the resist¬ 
ances per 1,000 ft. of the various 
sizes, B. &. S. gauge, of alumi¬ 
num wire multiply the values 
given in the sixth column of 


APPROXIMATE EFFI¬ 
CIENCIES OF DIRECT- 
CURRENT MOTORS 


Size of 

Efficiency 

of Motor 

of Motor 

Horsepower 

Per Cent. 

1 to 3 

80 

3 to 5 

82 

5 to 20 

85 

Over 20 

90 


the table by If. To obtain 
the carrying capacities of alu¬ 
minum w r ires, multiply the 
values in the seventh or eighth 
column or by .84. By divi¬ 
ding the values of the fourth 
or fifth column of the table by 
2 the approximate weights per 
1,000 ft. or per mi. of the 
various sizes of aluminum wire 
will be obtained. 

Wire Formulas.—The size of wire for a direct-current motor 
may be calculated by means of the formula 

20,470 H. P. L 
A = ---. 

E e n 

in which A =the size of wire, in circular mils; 

H. P. = output of motor, in horsepower; 

L = distance from the motor to feeding point, in feet; 
E = voltage on nameplate of motor; 
e = allowable drop from feeding point to motor; 
n — efficiency of motor. 

This formula allows for 25% overload. The value of n is 
rarely known, but may be taken from the accompanying 
table: 

The size of wire for induction motor circuits may be found 
by using the formula 


in which 


KU.P.L 

A =- 

Een 

K = constant; 









ELECTRICITY 


163 


and the remaining symbols have the same significance as 
in the formula for direct-current motors. For single-phase 
motors, the value of K may be taken as 47,560; for two- and 
three-phase motors as 23,780. Single-phase motors are rarely 
used in mines. The value of n for polyphase motors may be 
taken from the accompanying table of efficiencies. 

The amperes per terminal 
of an induction motor is often 
given on the name plate, in 
which case the following formula 
may be used: 

25.5 CL 
A — t 

e 

in which C = current rating; 
and the other letters have the 
meanings given before. 

It is important that induc¬ 
tion motors be operated at the 
voltage given on the name plate; 
therefore, in order to obtain 
the most accurate results, the value of e in the foregoing formu¬ 
las should be estimated as closely as possible. 

WIRING CALCULATIONS FOR LAMPS 

Incandescent-Lamp Ratings. —There are two classes of 
incandescent lamps in common use; the carbon lamp and the 
Mazda, or tungsten, lamp. Before the advent of the Mazda 
lamp, it was the practice to specify lamps by their candle- 
powers; as, for instance, a 16-candle power lamp or a 32-candle- 
power lamp, abbreviated to 16-c.p. or 32-c.p. Now, however, 
lamps are rated according to the watts required to operate them. 
A 16-c.p. carbon lamp is now known as a 50-watt lamp; a 
32-c.p. lamp becomes a 100-watt lamp. The Mazda lamps 
most commonly used are the 25-, 40-, 60-, and 100-watt sizes. 
Most incandescent lamps are made for voltages ranging from 
100 to 130, although lamps designed for other voltages may be 
obtained. 

Formulas for Lamp Wiring. —Because, with slight modifica¬ 
tions, it may be adapted to any of the common distribution 


APPROXIMATE EFFI¬ 
CIENCIES OF INDUC¬ 
TION MOTORS 


Size of 

Efficiency 

Motor 

of Motor 

Horsepower 

Per Cent. 

Under 5 

80 

5 to 20 

85 

20 to 100 

88 

Over 100 

90 









164 


ELECTRICITY 


systems, the most convenient formula for lamp-wiring cal¬ 
culations is 


KWL 



in which A =size of wire, in circular mils; 

K = constant; 

W = power required by all lamps of group to which 
wiring is run, in watts; 

L = distance from feeding point to lamps, in feet; 

E = voltage of circuit at lamps; 
e = allowable drop in circuit. 

For direct-current, two-wire and three-wire circuits and for 
single-phase, alternating current, two-wire and three-wire cir¬ 
cuits, the value of K is 22. For two-phase four-wire, and 
three-phase three-wire circuits, the value of K is 11. On a 
twoqihase or three-phase circuit, the lamps should be, as far 
as possible, distributed equally; that is, each phase should 
have the same number of lamps connected to it. 

If all the lamps are of equal size, W = Nw, where N is the 
number of lamps and w is the watts per lamp. The formula 
just given then becomes 

KNwL 

A =- 

Ee 

If a certain size of wire is at hand and it is desired to know 

the number of lamps that the given size of wire will feed with 

a certain drop, the last formula may be used in this manner: 

AEe 

N = - 

KwL 

The length to be used in the wiring formula is the average 
distance traversed by the current in the conductor. For 
example, if, as in view (a) the lamps were all grouped or bunched 
at the end of the line, the length used in the formula would be 
that from G to A. In case the load is uniformly distributed 
all along the line, as shown in view ( b ), the current decreases 
step by step from the dynamo to the end. In such a case, 
the length or distance to be used in the formula is one-half 
that used in the former case, or one-half the distance from the 
dynamo to the end. 



ELECTRICITY 


165 


Arc Lamps. —Arc lamps are frequently run on constant- 
potential circuits, and usually consume from 400 to 500 watts. 
There are so many types of these lamps that it is difficult to 



give any current estimates that will be generally applicable. 
Enclosed-arc lamps usually take from 3 to 5 amp. when run 
on 110-volt circuits. 


ELECTRIC SIGNALING 


BATTERIES 

Batteries are used for various purposes in connection with 
mining work, principally for the operation of bells and signals. 
The unit of a battery is called a cell, a battery consisting of 
several cells properly connected together. An ordinary cell 
consists, fundamentally, of two pieces of solid conducting 
substances in a liquid that acts chemically upon one more than 
upon the other. The liquid is called the electrolyte; the solid 
substances, electrodes. The electrode at which the current 
leaves the cell is called the cathode; the electrode at which the 
current enters is called the anode. The anode is usually made 
of zinc; the cathode of carbon, copper or iron. Most cells use 
a chemical depolarizer to remove the objectional hydrogen 
gas that is formed by the chemical action of the electrolyte 
on the anode. An open-circuit cell is one designed for work 
where current is used only intermittently and the circuit is 
usually open. A closed-circuit cell is one designed for circuits 
that are usually closed and the current passes continuously. 























166 


ELECTRICITY 


For operating electric bells, any good type of open-circuit 
battery may be used. The Leclanch6 cell is largely used for 
this purpose, also several types of dry cells. 

BELL WIRING 

The simple bell circuit is shown in Fig. 1, where p is the 
push button, b the bell, and c the cells of the battery connected 



up in series. When two or more bells are to be rung from one 
push button, they may be joined up in parallel across the 



Fig. 2 


battery wires, as at a and b. Fig. 2, or they may be arranged 
in series, as in Fig. 3. The battery B is indicated in each dia¬ 



gram by short parallel lines, this being the conventional method. 
In the parallel arrangement, the bells are independent of each 
























ELECTRICITY 


167 


other, and the failure of one to ring would not affect the other; 
but in the series grouping, all but one bell must be changed 
to a single-stroke action, 
so that each impulse of 
current will produce only 
one movement of the ham¬ 
mer. The current is then 
interrupted by the vibrator 
in the remaining bell, the 
result being that each bell 
will ring with full power. 

The only change necessary 
to produce this effect is to 
cut out the circuit-breaker 
on all but one bell by con¬ 
necting the ends of the mag¬ 
net wires directly to the terminals of the bells in the circuit. 

When it is desired to ring a bell from one of two places 
some distance apart, the wires may be run as shown in Fig. 4. 
The pushes p are located at the required points, and the battery 
and bell are put in series with each other across the wires 
joining the pushes. 

A single wire may be used to ring signal bells at each end of 
a line, the connections then being as shown in Fig. 5. Two 
batteries B and B', and a key and bell at each station are 
required. The keys are of the double-contact type, making 
connections normally between the bell and line wire L. When 
one key is depressed, a current from the battery B passes along 




the wire through the upper contact of key k' to bell b' and back 
through ground plates G' and G. 























168 


PROSPECTING 


When a bell is intended for use as an alarm apparatus, a 
constant-ringing attachment may be introduced, which closes 
the bell circuit through an extra wire as soon as the trip at 
door or window is disturbed. In the diagram, Fig. 6, the 

main circuit, when the 
push p is depressed, is 
through the automatic 
drop d by way of the 
terminals a and b to the 
bell and battery. This 
current releases a piv¬ 
oted arm which, on fall¬ 
ing, completes the cir¬ 
cuit between b and c, 
establishing a new path for the current by way of e, indepen¬ 
dent of the push p. 

The failure of a bell to operate is usually due to one of the 
following causes: Break in the circuit, crossed wires, wrong 
connections, weak battery, wrong adjustment of the vibrator. 
Inspection is necessary to locate the fault. If the bell rings 
weakly, it is usually due to poor insulation somewhere on the 
circuit or to a weak battery. Probably the most convenient 
way to test a battery for weakness is to connect a good bell 
directly across the battery terminals. 



PROSPECTING 

Formations Likely to Contain Coal. —No coal beds of impor¬ 
tance have as yet been found below the Carboniferous period. 
Though coal may be looked for in any later stratified or sedi¬ 
mentary rocks the bulk of the best coal has been found in depos¬ 
its made during this period. As a rule, highly metamorphic 
regions and regions composed of volcanic or igneous rocks 
contain no coal. The rocks most common in coal measures 
are sandstones, limestones, shale, conglomerates, fireclays, and, 
in some localities, beds of iron ore. 

An examination of the fossils contained in the rocks of any 
locality will usually determine whether the rocks belong to a 













PROSPECTING 


169 


1 


Cras 


American Periods, foreign Gfuivaienis. 


X 

$ 


Peceni 

Ghampiain 

Giaciai 


Peceni 

Pieisiocene 


Pg/eofMan 


Pi/ocene 


Pliocene 


% 



Miocene 


Miocene 


dye of 
Mammais 


Eocene 


Eocene 


i 

4 

5 


(i aramie Series) 
Upper Creiaceoi/s 


Upper Creicrceoe/3 






i ewerGreiaceous 
(Pa/roia Group) 
(Comanche Group) 


lon'erCrefaceeus 
or 

Peocomian 


3 

I 


dianiosaurus 

Peds 


Oo/iie 

Has 


dye of 
Pepiiies 


t 

•§ 

£ 


Gonneci/cuiPder 
0eds 


ifeuper&Phaei/c 
Musche/Paifc 
PunierGandsie/n 


Garhon/ferous 

or 

Cedi Measures 



Permian 

CarOen/fereus 

or 

GoaiMeasures 

Mounia/n 

f/mesione 


O/dPed 

Jandsione 


dyeof 
dmphih/ans 


or 


dyeof 
dcroyens 
fPianis ofihe 
CoaiPer/od) 


dye of 
C/sfes 



































































































































170 


PROSPECTING 


period below or above the Carboniferous, and hence whether 
there is a probability of the formations containing coal. There¬ 
fore, the prospector should familiarize himself with the geo¬ 
logical periods, and with the most common fossils of the various 
periods. The accompanying table gives the American and 
foreign names of the various geological periods, together with 
the name of the principal form of life during each period to 
and including the Devonian, which is the one below the Car¬ 
boniferous or coal-forming period. 

Coal or Bedded Materials. —The presence of the outcrop of 
any bed may often be located by a terrace caused by the dif¬ 
ference in the hardness of the strata, though any soft material 
overlaying a hard material will form a terrace. Usually, the 
outcrop of a coal terrace is accompanied by springs carrying 
iron in solution, which is deposited as ochery films upon 
the stones and vegetable matter over which the water flows. 
Sometimes the outcrop is characterized by a marked difference 
in the vegetation, for instance, the outcrop of a coal bed con¬ 
tained between very hard rocks will have more luxuriant 
vegetation than the surrounding country. 

Some indication as to the dip and strike of the material 
composing the bed may be obtained by examining the terrace 
and noting the deflections from a straight line caused by the 
changes in contour of the ground. Where a bed or seam is 
faulted, its continuation can frequently be found by breaking 
through into the measures beyond, when an examination of 
the formation will indicate whether the rocks are those that 
usually occur above or below the desired seam. 

Underground Prospecting. —Frequently a seam or deposit 
becomes faulted or pinched out underground, so that it is 
necessary to continue the search by means of underground 
prospecting. This work is, to a large extent, similar to surface 
prospecting, the underground exposures being simply additional 
faces for the guidance of the engineer. In the case of coal 
beds or similar seams, the manner of carrying on the search 
will depend on the character of the fault. Where sand faults 
or washouts are encountered, the drift or entry should be 
driven forwards at the angle of the seam until the continuation 
of the formation is encountered, when a little examination of 


PROSPECTING 


171 


the rocks will indicate whether they are the underlying or 
overlying measures. In the case of dislocations or throws, 
the continuation of the bed may be looked for by Schmidt’s 
law of faults, which is as follows: 

Law .—Always follow the direction of the greatest angle. 

It has been discovered that, in the majority of cases, the 
hanging-wall portion of the fault has moved down, and there¬ 
fore such faults are commonly called normal faults. For 
instance, if the bed a b were being worked from a toward the 
fault, upon encountering the fault, work would be continued 
down on the farther side of the fault toward d, until the con¬ 
tinuation of the bed toward b was encountered. In like manner, 
had the work been proceeding from b, the exploration would 
have been carried up in the direction of the greatest angle 
and the continuation toward a thus discovered. A reverse 
fault is one in which the 
movement has been in the 
opposite direction to a nor¬ 
mal fault. 

Exploration by Drilling 
or Bore Holes. —When test¬ 
ing soil or searching for pla¬ 
cer gold, sand, soft iron, or 
manganese ores, and similar 
materials that usually occur 
comparatively near the sur¬ 
face, earth augers, which are 
very simple in construction, may be used to great advantage. 

Percussion, or churn, drills are frequently used in drilling for 
oil, water, or gas, and were formerly much used in searching 
for coal and ores. They are at present little used in prospecting 
for either ore or coal, however, because they reduce the material 
passed through to small pieces or mud, and so do not produce 
a fair sample; besides that they can only drill perpendicular 
holes. 

The diamond drill is the only form that has been universally 
successful in drilling in any direction through hard, soft, or 
variable material. Even in the use of the diamond drill, many 
difficulties present themselves, and demand careful study in 
























172 


MINING 


adapting the form of apparatus to the work in Hand, and in 
rightly interpreting the results obtained from any set of obser¬ 
vations. The diamond drill is used very extensively and gives 
the best results of any kind of drilling for exploration. The 
work is best done by firms that do this special work and 
the surveying of the bore holes must be done by the mining 
engineer. 


MINING 


OPENING A MINE 

CHOOSING THE LOCATION 

The location of the surface plant and the mine opening 
depend on the formation of the deposit and on the facilities for 
transporting the product to market. It is impossible for one 
not on the ground, and unfamiliar with natural or railroad 
transportation facilities in the neighborhood, to give an idea 
regarding the second consideration. In regard to the first con¬ 
sideration, the following points should be observed: 

When the seam or vein outcrops within the limits of the 
property and is flat, a water-level drift is the best method of 
opening it. If the vein has any considerable inclination, it 
should be opened by a slope, or by a tunnel driven across the 
intervening measures. Where the deposit has an inclination 
of but from 1° to 1.25°, the water-level drift is generally used, 
and the main-haulage entry is opened at the lowest accessible 
point on the outcrop, which insures free drainage and a favor¬ 
able grade for haulage. Water-level drifts, however, are only 
profitable where the inclined seam is exposed in ravines or 
gorges eroded across the strike of the measure, or where the 
vein can be reached by a short tunnel from the surface to the 
seam across the measures. This is often the case when the 
seam dips with the hill, but when the dip is against the hill, 
the tunnel is generally a long one. While the expense of oper¬ 
ating a mine opened by a long tunnel is less than one opened 
by a slope or shaft, owing to cheaper drainage and haulage, 



MINING 


173 


when the coal above the water level is exhausted the tunnel 
is almost worthless. 

When the outcrop dips into the hill, the drift is usually 
commenced a few feet below the outcrop terrace, and is driven 
on a slight up grade until the normal dip is reached. When the 
inward dip is too strong, the better plan is to sink a shaft in the 
center of the basin, provided the depth is not too great and the 
amount of water to be pumped is comparatively small. If the 
inward dip to the center of the basin does not exceed a total of 
25 ft. difference in level, a drift may be used and drainage 
effected by a siphon. 

When the seam is inclined and is accessible at no point along 
its outcrop low enough to furnish sufficient lift or breast length, it 
should be opened by a slope or shaft. Or, if the seam is flat and 
does not crop on the tract, a shaft is the only method of working 
it, unless it lies so near the surface that it can be stripped. 

Where a seam has a dip of 20° or more, and is brought close 
to the surface by an anticlinal axis or saddle, a rock slope, or, 
in other words, a tunnel dipping the same as the seam may be 
started from the surface, and, when the seam is reached, may 
be continued to the desired depth in the seam. 

In sinking slopes for coal mines, it is customary to sink an 
airway alongside of and parallel with the slope, with a pillar 
of about 10 yd. between. The slope for coal mines is usually 
sunk so that there is a lift of from 100 to 110 yd., and then 
gangways are turned off on each side. The lift is the length on 
pitch that breasts or rooms, driven at right angles to the gang¬ 
way, can be driven in good coal. Subsequent lifts are usually 
from 80 to 100 yd. long. 


SHAFTS 

Shafts and tunnels may be temporary, or those that are 
simply driven for exploration purposes, and are not to be used 
for any great length of time; or they may be permanent, or those 
that are driven for a specific purpose and usually have a definite 
predetermined capacity. In the United States, shafts are 
usually square or rectangular in form, as timber is used in 
lining them. In Europe, round or oval shafts are frequently 
used; these have a lining of brick, iron, concrete, or masonry. 


174 


MINING 


Compartments.—The number of compartments in a shaft 
and their arrangement depends largely on the use to which the 
shaft is to be put; also on the number of shafts at the property, 
and the depth of the shaft. Where the material is to be 
removed is comparatively near the surface, it is usually cheaper 
to sink a number of two- or three-compartment shafts than it is 
to tram all the ore to one large shaft; while, in the case of very 
deep mines, large four- or six-compartment shafts are sunk, and 
the underground haulage extended over a greater area. When 
the shafts are lined with timber, a strong construction can be 
obtained by placing the compartments side by side, as shown 

in Fig. 1. When a body of mate- 
rial comparatively near the surface 
is being removed through a number 

---- of shafts, two-compartment shafts 

Fig. 1 are frequently built, both compart¬ 

ments being used for hoisting, and 
separate shafts being provided for the pump column and lad- 
derways. This reduces both the size of the shaft and the 
timbering necessary, and also does away with the special dan¬ 
ger from fire that always exists when there is a ladderway in 
the shaft, for it is always difficult to fight fire in these special 
compartments. 

Size of Shafts.—Shafts vary greatly in size, depending on 
the number of compartments desired and the size of the com¬ 
partments. For coal mines, they are generally from 10 to 12 
ft. wide inside of timbers, and each compartment is from 6 to 7 
ft. wide inside the guides. This would make the outside 
dimensions of a double-compartment shaft about 13 to 15 ft. 
wide, 17 to 18 ft. long, and a triple-compartment shaft from 
24 to 25 ft. long. 

Shaft Sinking.—As a general thing, the loose material or 
wash above bed rock is not thick enough to cause any serious 
trouble, and ordinary cribbing of heavy timber or a masonry 
curbing is sufficient. But when the surface is very thick or 
loose, and runs like quicksand, considerable difficulty is expe¬ 
rienced. The general method of overcoming this difficulty in 
the past was to at once divide the shaft into the required number 
of compartments by heavy timbers alternating or placed skin 






MINING 


175 


to skin, which had the effect of bracing the cribbing against 
the lateral pressure of the loose material. This method is 
effectual where the wash will remain solid or stand long enough 
to allow the timbering and cribbing to be put in. But when the 
surface is thick, loose, or watery, or of quicksand, some one of 
the following special methods of sinking must be adopted: 

Forepoling 

Metal linings, forced down with¬ 
out use of compressed air 
Pneumatic method, limited to 
about 100 ft. in depth 
Poetsch process, freezing method 


Quicksand. 


Rock, hard or soft, but very 

wet . Kind-Chaudron method 

Rock, hard or soft, but not 

very wet. Continuous, or long-hole, method 

When the ground is so bad that it will not stand for several 
days between excavation and the completion of the lining, it 
becomes necessary to carry the timber to the bottom of the 
work. This may be accomplished by using square-set shaft 
timbering and driving laths, or forepoling behind the timber so 
as to keep the soft material from running into the opening. 
The advantages of forepoling are that, if the shaft is being lined 
with square sets, forepoling can be commenced at any point, 
and, if the ground is not too bad, the work can be continued 
by this means until solid material is encountered. When the 
ground is particularly bad, it may become necessary to use 
breast boards, which are simply boards braced against the bottom 
of the shaft so as to keep the material from rising into the open¬ 
ing, only one board at a time being removed while the material 
behind it is excavated. 

The pneumatic method of shaft sinking was developed from the 
system in use for putting down foundations for bridge piers. 
At the bottom of the shaft there is a small chamber called a 
caisson, in which a sufficient air pressure is maintained to 
exclude the water at all times. The shaft lining is built on 
above this chamber, and gradually forced down into the soil. 
Men enter the chamber and excavate the material from under 
the caisson as it descends. 






176 


MINING 


By this method the sinking commences at once and is contin¬ 
ued without interruption until the lining is completed to bed 
rock, to which the lining is joined, as shown in Fig. 2. An air 
compressor, which is subsequently used, is the only auxiliary 
machine necessary. 

In the pneumatic process, the fine material is aspirated out 
of the caisson by the air pressure. This process is limited to a 
depth of about 100 ft., as it is impossible for men to work under 
a greater air pressure than that which corresponds to about 100 

ft. of hydrostatic pressure. 

By the freezing process, 
pipes are sunk in the ground 
about the area to be frozen, 
as a rule, not more than 3 
or 4 ft. apart. The lower 
ends of the pipes are sealed 
or closed, and an inner tube 
introduced so that a freezing 
mixture may be caused to 
circulate down through the 
inner tube, and up through 
the outer tube. This free¬ 
zing mixture may be liquid 
ammonia gas, which is al¬ 
lowed to expand in the outer 
tube, or a solution of cal¬ 
cium chloride that has been 
reduced to a very low tem¬ 
perature by means of an 
ordinary refrigerating machine. The circulation is maintained 
in the pipes until the ground between them is frozen solid, 
after which the work may be continued as though the forma¬ 
tion were solid rock, the material being blasted and hoisted 
in buckets. The freezing process may be applied to any 
wet formation, whether hard or soft, while the pneumatic 
process is applicable only to soft formations. The free¬ 
zing process may be carried to practically any depth. 
As a rule, the freezing pipes are never sunk inside of the 
shaft area. , 








































MINING 


177 


The Kind-Chaudron method is applicable only to round shafts, 
and is suitable for shafts passing through very wet and at the 
same time comparatively soft formations. The excavation is 
carried on by means of a large set of boring tools armed with 
steel teeth, and operated in a manner similar to that employed 
in drilling wells by the percussive system. 

The long-hole process consists in the drilling of a series of 
diamond-drill holes over the area of the proposed shaft, then 
filling the holes with sand. Afterwards 5 or 6 ft. of sand is 
removed from the holes in the interior of the shaft, and these 
holes are charged with explosives, and fired by electricity. 
Next, the holes around the boundary of the shaft are charged 
and fired in the same manner, and the process is continued until 
the bottoms of the diamond-drill holes are reached. This 
method is especially applicable to work in hard rock, where 
great speed in sinking is desired, for all the drilling is accom¬ 
plished at one operation, after which the sinking progresses by 
simply cleaning out the drill hole, blasting the material and 
cleaning it away. 

Sinking Head-Frames. —Head-frames of very simple form 
are used for sinking. The skeleton of the frame is formed of 
heavy squared timber 10 in.XlO in. or 12 in. X 12 in. mortised 
and pinned together, and braced by diagonal braces. A good 
height from the surface to the center of the sheave is from 20 to 
25 ft. The sheave should be from 6 to 8 ft. in diameter. 

Sinking Bucket. —The sinking bucket should be of boiler 
iron, or of heavy hardwood strengthened by iron bands, about 
3 ft. in diameter at the top by from 2£ to 3 ft. deep.- It should 
be suspended by a handle pivoted a trifle below the center, and 
it should have a pin on the rim of the bucket that will hold it in 
an upright position when a loose ring on the handle is slipped 
over it. A chain fastened to the top of the head-frame, with 
a hook on its loose end, is suspended so that, when hanging 
plumb, it is over a chute leading to the dump car. As the 
bucket is hoisted out of the shaft, this chain is attached, and the 
engine reversed. The bucket swings over the chute, the ring 
holding it upright is knocked off the pin, and the rock is dropped 
into the chute. Rocks too large for the bucket are suspended 
in chains and are hoisted in that way, and removed on a truck 



178 


MINING 


that runs on a track inside of the head-frame, the gauge of 
which is sufficiently wide to give plenty of clearance for the 
bucket. 

Sinking Engines.—Most shafts and slopes are sunk with 
old engines, or else by engines especially designed for such work, 
and so constructed that they can easily be moved from place 
to place. 

Tools.—The old method of hand drilling is still adhered to 
in many instances, but it is gradually giving way to machine 
drilling, especially in deep shafts. When properly managed, 
the work is done much more rapidly and economically by the 
several excellent types of rock drills now on the market. They 
are constructed in a variety of shapes by the makers, and there 
are so many convenient accessories in the shape of fittings, etc., 
that all contractors prominent in the various coal fields possess 
one or more of their favorite type of drills. These drills are run 
either by compressed air, steam, or electric power, and in large 
shafts two are usually employed, so that work may not be 
delayed by a breakdown of one drill. 

Drainage and Ventilation.—When only a small amount of 
water is encountered while sinking, the best plan is to allow 
it to collect in a depression and bail it from there into the bucket, 
hoisting it the same as the rock. Where the water "is excessive, 
in quantity, a steam pump is necessary; all the leading pump 
works make pumps especially designed for sinking purposes. 

When the shaft is of moderate depth, a fire burning in one 
corner will supply ample ventilation. To rapidly clear away 
smoke, a good plan is to bum a bundle of straw or shavings in 
one end of the shaft, and throw a couple of buckets of water 
down the other end. When the shaft is very deep, or when the 
sectional area is small, ventilation is produced either by a steam 
jet, or by a small fan turned either by steam or by hand. In 
some cases, a fire is used that draws into a board pipe. 

Slope Sinking.—A slope is an inclined plane driven down on 
the bed of the seam, and is generally through coal or ore, though 
sometimes it is driven through rock across measures to cut the 
seam that cannot be conveniently worked by a slope. In the 
latter case, it is merely an inclined tunnel; in the former it 
might be termed an inclined gangway. A slope and an inclined 


MINING 179 

plane, when mentioned hereafter, will mean an inclined opening 
in coal used as a passageway for mine cars. 

When the location of the slope has been decided on, a tem¬ 
porary sinking plant is erected. For a short distance, varying 
with the nature of the ground, but usually ranging from 10 to 
20 ft. on the pitch, an open cut is made, and the earth, rock, or 
crop coal is thrown out by hand. As soon as sufficient cover is 
reached, the work of undermining and timbering is commenced, 
and at the same time a double or single track is laid, so that the 
material can be taken out in a car or self-dumping skip. When 
the latter is used, the track is continued up a trestle some dis¬ 
tance above the surface, and a head-sheave so placed as to draw 
the skip up the required distance and dump the material in a 
chute beneath the trestling. 

The width of the slope depends on the size of the cars and the 
number of compartments. The most common arrangement is 
to divide the slope into three compartments; two large ones 
for hoistways, and a smaller one for pump rod, column pipe, 
steam pipe, and traveling way. This last is also used as an 
airway while sinking is going on. 

The Sump.—When the shaft or slope is completed, among the 
first things necessary is a sump in which to collect the drainage 
of the mine. This is an opening lower in the vein, when it is 
a pitching one, or in the rock when it is a flat seam reached by a 
shaft. It should be large enough to hold any excess of water 
that the pumps cannot handle; and the pumping machinery 
should be powerful enough to handle the ordinary drainage by 
running not over 10 hr. per da. When this is the case, in 
an emergency, the pumps can be run continuously, and thus 
handle the surplus water. 

Driving the Gangway.—In bituminous coal seams, the height 
of the gangway is governed by the thickness of the seam; this 
is also true, in a certain sense, in the anthracite regions, though 
in anthracite mines they are very seldom less than 6 ft. in 
height. In the larger seams they are from 6 ft. 6 in. to 7 ft. 
6 in. high in the clear, and from 10 to 15 ft. wide. The gauge 
of track varies from 24 to 48 in. The grade should rise at least 
4 in. in 100 ft., and a gutter 3 ft. wide by 18 in. deep should be 
cut in the coal on the low side. 


180 


MINING 


TUNNELS 

Mining tunnels are usually of small cross-section compared 
with those that occur in railroad work, it being rare that their 
size is such that they cannot be driven in full section, and if the 
ground is firm the operation of placing the lining may follow 
behind the work of driving. They are generally lined with tim¬ 
ber, and in case the ground is of a soft or treacherous nature, 
bridged square sets and forepoling are employed, with or with¬ 
out breast boards, as the necessity of the case demands. When 
the material is firm rock, the tunnel is sometimes not lined, the 
roof being given an arched form. 


MINE TIMBER AND TIMBERING 

Choice of Timber. —Timber used for underground supports 
in mines should be long-grained and elastic, and, at the same 
time, should not be too heavy. Oak, beech, and similar woods 
are very strong, but are heavy to handle, and when set in place 
are treacherous, because they are short-grained and not elastic, 
so that they break without warning. Mine timber is placed, 
not with the intention of ultimately resisting the great pressure 
of the earth, but to keep any loose pieces in place and to give 
warning to the workmen, thus enabling them to escape before 
a fall occurs. For this reason, pine and fir are suitable for mine 
timbering, as they combine a fair amount of strength with con¬ 
siderable elasticity, and hence give warning long before they 
break. Very elastic timbers, such as cypress, willow, etc., are 
to be avoided, for they simply bend like a bow and do not offer 
the necessary resistance to hold the material in place for a short 
time. 

Preservation of Timbers. —The character of the ventilation 
in a mine has considerable effect on the life of any timber sup¬ 
ports. Damp stagnant air will cause mold and fungus growth, 
which will be followed by the destruction of the timber through 
decay or dry rot. All timbered openings should be well venti¬ 
lated, and provision made for the speedy removal of damp hot 
air, such as commonly occurs around pump rooms and along 
steam lines. Water is a good preservative, as it washes off the 



MINING 181 

spores of the fungi as fast as they are formed, and for this reason 
shaft timbers are sometimes kept wet. 

Timber may be also preserved: (1) by a solution of common 
salt and water; (2) by impregnating the wood with such metallic 
substances as sulphates of copper, iron, etc.; (3) by impregna¬ 
tion with the chloride of magnesium or zinc; (4) by creosoting; 
(5) by coal tar; (6) by carbolineum. 

A solution of 1 lb. of salt in 4 or 5 gal. of water gives a cheap 
and easily applied preservative with which the timber should 
be thoroughly soaked. Sulphate of iron is economical and 
effective. In the zinc process, a solution of 1 gal. of liquid 
chloride of zinc (sp. gr. 1.5) mixed with 35 gal. of water is 
forced into the wood by pressure. Impregnation with crude 
creosote oil is effective, for the creosote fills the pores and pre¬ 
vents saturation by water; it destroys organic life; the carbolic 
acid that it contains coagulates the albuminoids and prevents 
decay; but it has the disadvantage of making the timber very 
inflammable. Painting with liquid tar is effective, but makes 
the wood very inflammable. Painting with ordinary white¬ 
wash is also said to give good results. Carbolineum is said to 
be effective, but is quite expensive. It is applied with a brush, 
or by steeping in a tank; 1 gal. will cover 300 to 400 ft. of tim¬ 
ber. It has been shown that preservatives decrease the strength 
of timber from 8% to 20%, depending on the process used. 

In selecting props, the principal points to be observed are: 
Straightness, slowness of growth as indicated by narrow 
annular rings, freedom from knots, indents, resin, gum, and sap. 
They should also be well seasoned before use. With these 
precautions and proper mine ventilation, fungus growth may 
generally be obviated and durability insured. 

Placing of Timber. —The individual sticks should never be 
weakened by cutting mortise and tenon joints. The pressure 
should be evenly distributed over a number of sticks, and not 
concentrated or centered at one point. Centers of revolution 
should be avoided. The individual sticks should be placed in 
the direction of the strain that they are to resist, so that they 
will be subject to compression along their length rather than 
to a transverse strain. The individual sticks should be so 
placed, and the joints so formed, that the pressure tends to 



182 


MINING 


strengthen rather than weaken the structure up to the crushing 
strength of the timber. In the case of large stopes, the timber¬ 
ing should be done according to some regular system, while, 
at the face of coal mines, single props or posts are usually better, 
owing to the fact that their duty is only to support the loose 
portion of the roof for a limited time. Probably the most 
important point is to timber in time, before the rock becomes 
broken or begins to settle. 

It seems generally agreed that the main weight in mines 
comes nearly at right angles to the bedding, and that the props 
should be mainly set in that direction. If the deposit is hori¬ 
zontal, the weight generally comes vertically; but if the deposit 
is inclined, the weight comes at a right angle to the inclination. 
Some authorities hold it as a principle that all props should be 
set parallel with the main pressure. Others, in order to guard 
against possible side thrusts and a tendency of the ordinary 
weight to ride to the dip in inclined deposits, purposely cause 
a sufficient number of props to be set with their tops slightly 
uphill. 

Sawyer fixes a maximum and minimum slope for the props, 
varying with the rate of dip. He makes this maximum slope 
of the props one-sixth that of the dip, and the minimum slope 
one-third of the one-sixth. 

Props are usually set with the butt end downwards, but not 
always. Placing the butt end upwards adds a trifle to the 
weight on the lower end, but the larger size at the top lessens 
the liability of its being split by a coupling resting on it, and 
also gives more surface for abrasion in hammering up against 
a rough roof. 

Joints in Mine Timbering.—In all mine timbering, the object 

is to so form the joints that no fastenings will be necessary and 
that the pressure from the surrounding material will keep the 
joints tight. The reason for this is that metal joints usually 
corrode rapidly in mines, while the timbering can be replaced 
with greater ease if the sticks are so framed that, by relieving 
them temporarily of the pressure from the sides and top, they 
can be simply lifted out of place and new ones substituted. 
The use of a framing machine renders it possible to frame the 
joints more exactly than with hand framing. With hand-framed 


1 


MINING 


J83 


timbers, the joints are always cut a little free to allow for 
any unevenness in the surface, but, if machine-framed, they 
are sure to be of the same size. As timber does not shrink in 
the direction of its grain, if the caps shrink slightly, they will 
become loose in the space between the shoulders; hence, if 
timbers are cut green and framed to the exact size, subsequent 
shrinking may open some of the joints. This may be obviated 
by keeping the timber moist. 

Undersetting of Props.—Props at the working face should 
not be set at right angles to the inclined floor of the seam, but 
should be underset, and the greater the inclination, the greater 
the underset. The amount of underset should vary with the 
inclination of the seam, and should not be so great that the 
props will fall out before the roof has tightened them. 

Forms of Mine Timbering and Underground Supports.—The 
timbering of a mine may be divided into two heads: timbering 
the working faces and timbering the roads. The roof may be 
supported (1) by packing the waste places entirely where 
sufficient material is obtainable for the purpose, and timbering 
the face and roads; (2) by partially packing the waste, by 
cribs or stone pillars with intervening spaces, and by tim¬ 
bering the face and roads; (3) by timbering the face and roads 
and supporting the roof in the waste places by wooden or stone 
pillars, but without any packing; (4) by timbering alone with¬ 
out any packs or walls whatever; (5) by supporting the 
main roads with brick arching, or by steel or iron supports. 
The accompanying figures show a number of the common 
forms of mine timbering and underground supports. 

Fig. 1 shows a post a and breast cap b. The breast cap b is 
also sometimes called cap, head-block, headboard, lid or bonnet. 
Sometimes the posts are placed upon blocks of wood similar 
to the head-blocks or headboards, the block being called a sole; 
at other times, two or more posts may be set upon one long 
block of timber called a sill. When posts are used in inclines, 
they should not be set perpendicular to the foot and hanging 
walls, but should be underset slightly, so that any tendency 
of the hanging wall to settle will bring the posts nearer at right 
angles to the walls, and so tighten them; the amount of under¬ 
set should never be more than one-sixth thej>itch of the deposit. 


184 


MINING 


Where posts are set at an angle, they are usually placed on 
wedges, and, as the pressure conies on, the wedges are tightened. 



Fig. 1 



Fig. 2 


Fig. 2 represents a stull a, which is used either to keep the 
walls of perpendicular or steeply inclined beds or veins apart, 
to support planking or lagging as a working platform, or as a 
platform upon which to pile ore or rock. 

Fig. 3 represents cockermegs , which are simply timber frames 
used in coal mines for holding the face of the coal in place while 
it is being undercut. They are composed of a pole c extending 
along the face and supported by short stulls or braces a, the 
whole being tightened into place by the long stulls b. 



Fig. 3 


Fig. 4 shows a crib, cog, chock, pillar, or shanty built up of 
timbers and filled with waste rock. It is intended to serve as a 

















MINING 


185 


pillar and to withstand great vertical pressure, doing away 
sometimes with the necessity of leaving pillars of ore. Fig. 5 



Fig. 4 Fig. 5 


is a cribbing framed from round timbers laid skin to skin, and 
used in raises or ore chutes. 

Gangway or Level Timbers. —Fig. 6 is a set used in the case of 
an extra-wide gangway, there being a center post under the 
middle of the cap. This form of set may be provided with a 
sill when the floor of the drift or gangway is soft. Fig. 7 shows, 
a form of drift set surrounded 
by bridging and used where 
such bad ground is encountered 
as to necessitate forepoling. 

At A are shown the posts, B the 
caps, and C the sill of the regular 

Up 

H 



Fig. 6 



set; D are upright bridge pieces; E a horizontal bridge piece 
separated from the set proper by blocks F so as to provide 




























































186 


MINING 


spaces H around the regular set through which the spiles or 
forepoles can be driven. Fig. 8 shows a form of drift set some¬ 
times used in very heavy or swelling ground. This method 




Fig. 9 


Fig. 8 



of framing the timbers shortens each piece and reduces the 
transverse strain on all the timbers. 

Fig. 9 shows an ordinary drift set provided with a sollar for 
ventilation purposes. An additional brace b is placed parallel 
to the cap c, and this is covered with plank lagging a, so as to 

provide a passage above 
the regular drift, which 
may be used as a return 
air-course. 

Fig. 10 is a simple 
form of drift set used 
when the roof and walls 
are of soft material, but 
the floor material firm. 
It is composed of posts 
l, upon which is placed 
the cap c. The joggle 
cut into the cap to receive the heads of the post should never be 
less than 1 in. nor more than one-third the thickness of the cap. 
The cap is usually made of such a length that the posts l have 

















MINING 


187 


an inclination or batter as shown in the illustration, thus giving 
greater strength to resist side pressure without decreasing the 
floor area of the drift, which may be necessary for drains, 
ditches, water pipes, etc. at the sides of the track. When the 
floor is not composed of solid material, the posts l may be set 
upon a sill that is framed to fit the legs in a manner similar to 
that shown for the cap. The joggle cut in the sill should never 
be less than 1 in. nor more than one-third the thickness of the 
sill. The sill is usually composed of lighter material than the 
cap, is flattened on one or both sides, and is sometimes used as 
one of the ties to receive the track. 



Fig. 11 


Fig. 11 shows a post l and the cap or collar c, used where one 
wall is of firm material. On one end the cap is placed in a 
hitch. When the collar is supported in a hitch, it is sometimes 
said to be needled, the operation being called needling. The 
bottom of the post a is also secured in a hitch, in case there is 
any side pressure. To keep the surrounding material in place, 
lagging is necessary, as shown behind the timbers in Figs. 10 
and 11. In the case of running ground, the lagging is usually 
made from sawed material and driven close together. 

Fig. 12 illustrates a method of spiling or forepoling; a are the 
posts of the regular set, b the caps, and e the top bridging. The 
front ends of the spiles from any given set rest on the bridging 








MINING 




Fig. 12 


of the next advanced set, and the spiles for advancing the work 
are driven between the bridging and the set, as shown in the 

illustration. To force 
the spiles out into the 
ground, so as to provide 
room for the placing of 
the next set, tail-pieces i 
are placed behind the 
back end of the spiles as 
they are being driven. 
After the spiles have 
been driven forwards the 
desired amount, another 
set is placed, the tail-pieces knocked out, and the front 
end of the spiles allowed to settle against the bridging of a 
new set. Where the face is composed of extremely bad 
material, it may be necessary to hold it in place with breast 
boards k, which are held in place by props l, that rest against 
the forward set. When breast boards are used, it is usually 
necessary to employ foot and collar braces between the sets, 
so as to transfer the pressure of the breast back through several 
sets. 

Fig. 13 shows a method of placing drift sets in the case of very 
heavy or swelling ground; a are the posts, b the caps, c the sills, 
<} the collar braces that bear 
against both the caps and 
the posts, e the foot or heel 
braces that bear against both 
the sills and the posts, / the 
diagonal braces that are 
halved together and placed 
as shown. 

Shaft Timbering. — Fig. 

14 shows square-set timber¬ 
ing, sometimes used for shaft 
lining; A are the wall plates, p IG jg 

B the end plates, C the 

buntons, and D the posts. The method of framing the different 
parts is plainly shown. 























































MINING 


189 


Fig. 15 represents cribbing sometimes used for shafts. It is 
composed of heavy sawed material halved together at the ends, 



Fig. 14 


as shown. The long pieces a are called wall -plates, and the 
short pieces b, end plates. Between the compartments a parti¬ 
tion is built up of pieces c called bunions. The ends of the bun- 
tons are let into the wall plates an inch or so, and should be 
so placed that they will break joints with the individual pieces 
of the wall plates, thus preventing the timbers of any single set 
from bulging into the shaft. 



Fig. 15 


Fig. 16 shows another method of framing, sometimes used 
for the end and wall plates where square-set timbering is used 



























































































































190 


MINING 


in shafts. The end and wall plates are halved together as 
shown. A beveled face is often formed at D. This construc¬ 
tion necessitates the cutting of a tenon on the end of the post F 


Wa/JP/afe 



Fig. 16 


as shown; 5 is a 2 in.X2 in. strip nailed along the center of 
the back of the wall and end plates as a support for the lagging 
that is placed outside of the sets. The lagging is usually com¬ 
posed of 2 in. X 3 in. plank. 

Fig. 17 shows the use of hangers between the individual 
square sets. The hangers are bolts provided with hooks on the 
ends, and are used to support the sets as the work progresses, 
the posts serving to keep the sets properly spaced, while the 

hangers keep the sets tight against 
the posts. Hangers are not always 
left in permanently, but may be 
removed after a considerable sec¬ 
tion of the shaft has been completed. 

Fig. 18 shows a method of apply¬ 
ing rough square sets, made from 
round timber, to the sinking of a 
small prospecting shaft by the use 
of forepoling; A is the first set of 
timbers and J is the second. The 
hangers are made from 2 in.X4 in. 
timbers F spiked to the sets and to the supports G. The supports 
G from which the sets are hung are placed over sills H , which are 
situated at a convenient distance from the collar of the shaft. 









































MINING 


191 




Fig. 18 


Fig. 19 
































192 


MINING 


At D is shown the lagging of the first set that is usually spiked 
to the set and at K the forepoling, which becomes the lagging 
between the second and third sets, and C the tail-pieces used 
for forcing the lagging out into the ground. The hangers 
between the next two sets would be spiked to the other two 
timbers of the sets. Where the bottom of the shaft is very bad, 
it may be necessary to use breast boards, as shown in Fig. 19, 
in which the shaft is being put down by means of square sets 
and forepoling with the use of breast boards. 



Square Sets.—Fig. 20 illustrates one method of framing 
square-set timbers from sawed material for use in stopes in 
mines; A are the posts, B the caps and sills, while C are the 
sprags or stuttles. The method of framing the joints is clearly 
shown in the illustration. Sometimes both caps and sprags are 
made of the same sized material and are framed alike. Fig. 21 
shows a method of framing round timbers for square sets. The 
dimensions / and c are usually made about 10 in., d, e, and i, 
each 2 in.; a depends on the diameter of the post that is to be 
used; and b as a general rule is cut down to an angle of about 45°. 





















































MINING 


193 - 



Landings, Plats, or Stations.—Fig. 
timbering a plat or 
station. The regular 
square-set timbering of 
the shaft is continued 
past the station and the 


22 is one method of 


Fig. 21 


Fig. 22 



heavy stull or reacher a put across at the bottom of the station.. 
The posts b are bolted against the posts of the sets and the; 


Fig. 23 


Fig, 24 


cap c placed on top of them. After this, the wall plates 
are cut out between the posts b, and the station opened and 








































































194 MINING 

timbered as shown. The height of the station is gradually 
reduced to that of the drift or level connecting with it. 

Fig. 23 shows a method of timbering a 
level in a slope where the ground is so 
firm that only stulls are used in the slope 
and at the station, the timbers all being 
secured in hitches or by stulls. Here 
a represents the stulls and c the tim¬ 
bers that are spiked to the stulls and 
carry the stringers for the car track; 
b represents the car track from the level 
that is brought across above the skip 
track. 

Special Forms of Supports.—Fig. 24 
shows a stone arch that, as a stull, sup¬ 
ports the waste material in the level. 
Fig. 25 shows a stone arch when one wall 
of the formation requires support. Fig. 26 illustrates a passage 
lined by a combination of stone or brick walls with wooden caps 
and lagging for the roof. Fig. 27 illustrates the lining of a drift 
or level supported by means of iron or steel shapes bent into 
the form of an arch and used for the support of lagging. 

Fig. 28 illustrates a cast-iron post or stull that has been 
successfully used as a support in mines. It is composed of two 



Fig. 25 



Fig. 26 



Fig. 27 


pieces a and b, held together by a collar c. By driving the 
collar c down on the post, the two pieces can be taken apart and 
the post moved. 
























MINING 


195 


Fig. 29 illustrates a masonry shaft lining supported by 
means of cast-iron plates C set in bell-shaped cavities cut 
in the walls of the shaft. As the masonry of that section 
from below is built up toward that above, the overhang¬ 
ing portion D is cut out a little at a time, and the 
masonry from below built up under the plate so that 
the lining becomes continuous. 

Fig. 30 illustrates masonry shaft linings, supported by 
artificial stone or cement foundations built in bell-shaped 
cavities cut in the walls of the shaft. The blocks of arti¬ 
ficial stone are provided with inclined bearings C, which 
serve to transmit a portion of the downward thrust of 
the lining in the direction of the arrow. 

Iron and Steel Supports.—The use of iron or steel, 
either for vertical or horizontal supports in mines, has 
not become at all general. In America, timber is as yet 
comparatively cheap in most mining localities, but this 
situation is fast changing and the timber reserves are 
being rapidly cut off, so that many mines now using 
wood must, in the comparatively near future, resort ^ IG ‘ ^ 

to some other form of support. 

Trestles.—Figs. 31 and 32 
illustrate the various timbers 
and methods of cutting the joints 





Fig. 29 


Fig. 30 


for ordinary railroad trestles. In Fig. 32 (a) is shown the 
manner of framing a pile trestle, while ( b ) represents the man¬ 
ner of placing timbers and cutting the joint for the framed 





















































196 


MINING 


trestle. Fig. 31 represents bents of a frame and pile trestle 
and the side elevation of a low pile trestle. Fig. 33 shows a bent 


Z 


Fig. 31 





of a framed trestle that is fastened together entirely by means of 
drift bolts, no joints whatever being cut. 



The various parts of the trestles shown are numbered; their 
names are as follows: 































































MINING 


197 


1 . 

2 . 

3. 

4. 

5. 

6 . 


Framed bent 
Pile bent 
Cap 

Cross-tie 

Gaining, dapping, or notching 
Guard-rail 



7. Jack-stringer Fig. 33 

8. Longitudinal brace or waling strip 


/ / 

9. 

Mortise 


l 'w*] 


/ / 

10. 

Mud-sill 



1 

¥ 

\ n. 

Packing block 



ill 

ill 

M US 

12. 

Packing bolts 

. L 

LJ 



FlG - 34 piles ’ 

14. Vertical, plumb, or upright piles 

15. Vertical, plumb, or upright posts 

16. Batter or inclined posts 

17. Sill 

18. Stringer 

19. Sway-brace 

20. Tenon 


Fig. 35 


Fig. 36 


Fig. 37 


Figs. 34 and 35 illustrate one manner of cutting the tenons 
and mortises on the ends of the batter braces and posts and 
frame bents, and also the drain holes that are bored in the mor¬ 
tise to prevent the timber from rotting. Usually the sills are 



a- 




Fig. 40 


Fig. 41 


notched or boxed to receive the ends of the timbers, in addition 
to having mortises formed in them. Figs. 36 and 37 show such 






























































198 


MINING 


joints for receiving the batter brace and post. Fig. 38 shows 
how a tenon is sometimes formed on the top of the pile to secure 
the cap. When the cap is secured by a tenon, the two are 
united by a wooden pin shown in the lower part of the figure, 
known as a treenail. 

Fig. 39 shows how the cap may be placed upon a pile trestle' 
by splitting the cap into two pieces, a and b with the tenon c 
the full width of the pile between them. Fig. 40 shows how 



Fig. 42 



Fig. 43 


the cap is sometimes secured to a pile by means of a drift bolt, 
and Fig. 41 shows how the same thing may be accomplished 
with the use of a dowel. 

Figs. 42 and 43 show two methods of longitudinal bracing 
between the bents of the trestles for inclined planes, such as are 
used at breakers or concentrating mills. Fig. 44 is an elevation 
of a high trestle, showing the cross-bracing and framing of the 
structure. 






































43 0 


MINTNG 


199 









































































200 


MINING 


Timber Head-Frames or Head-Gears.—Fig. 45 is the sim¬ 
plest form of head-gear. This consists of a vertical post, which 
carries the weight of the sheave, etc., and a diagonal post that 
approximately bisects the angle between the rope from the 
drum and the rope hanging down the shaft, thus taking the 
resultant pull upon the axle of the sheave. There is usually 
some extra timbering, as shown, to support the cage guides 

and form a platform 
about the sheave for 
convenience in oiling. 

Fig. 46 shows a modi¬ 
fied form of the same 
type of frame, in which 
the main upright leg is 
vertical and in which 
there is also another ver¬ 
tical member on the op¬ 
posite side of the shaft. 
The inclined leg is also 
braced and connected to 
the main vertical mem- 
F ig * 48 ber. Fig. 47 is a head- 

frame for an inclined shaft where the coal or ore pocket is in 
the structure carrying the sheaves. Such head-frames are some¬ 
times enclosed in their upper portions so as to protect the men 
during winter. Fig. 48 is a form of framing quite common in 
the anthracite fields of Pennsylvania, in which the timbers are 
further braced by tie-rods, as shown. 



METHODS OF WORKING 

GENERAL DESCRIPTION 

No definite rules can be given for the selection of a method of 
mining that will cover all the conditions that may exist at any 
given mine. Each mine is a distinct and separate proposition, 
and each superintendent must adapt the general principles here 
given to the local conditions at his own mine. Every system 
of mining aims to extract the maximum amount of the deposit 



































































MINING 


201 

in the best marketable shape and at a minimum cost and 
danger. 

The elementary conditions affecting the extraction of coal 
are: (1) weight of overlying strata or depth of deposit, (2) 
strength and character of roof, (3) character of floor, (4) 
texture of bedded material, (5) inclination and thickness of bed, 
(6) presence of gas in seam or in adjoining strata. 

Open Work.—Open work applies to the working of all 
deposits that have no overburden, or to those in which the 
overburden or overlying material is stripped from the portion 
of the deposit to be removed by hand, steam shovels, scrapers, 
etc. It includes particularly all quarries and placer workings, 
and can be applied to many mineral and coal deposits. Open 
work may be divided into two general classes: Where the 
w'hole or a greater part of the deposit is of value and has to be 
removed, as in quarries and in ordinary mines; where the val¬ 
uable portion is but a small part of the whole, as in placers or 
fragmental deposits carrying gold, platinum, etc. 

Closed Work.—Under the heading of closed work, it is 
customary to divide the methods of mining into coal-mining 
methods and metal-mining methods. This classification, how¬ 
ever, is not entirely logical, for identical methods are applied 
regardless of the mineral. A more logical classification is one 
based on the position, character, and thickness of the deposit, 
but the older classification has become so firmly established that 
it is not advisable to disregard it entirely. 

The typical and most extensive bedded mineral deposits 
are of coal and iron ore, and of these the former is by far the 
more extensively mined. A description of the several methods 
of mining coal beds will therefore comprise not only all the 
essential points in the mining of other bedded deposits, but will 
include a number of points not usually considered. The chief 
of these is the presence of explosive gas in such quantities as to 
influence the choice of a method of mining. 

The panel system divides a mine into districts or panels by 
driving entries and cross-entries so as to intersect one another 
at regular intervals of, usually, about 100 yd. Large pillars 
are left surrounding the workings within each panel, and any 
method of development may be used for each panel. This 


202 


MINING 


system has the following advantages: Better control of the 
ventilation, because the air in any panel may be temporarily 
increased or decreased, as required; an explosion occurring 
in one panel is less liable to affect the other workings. Coal 
may be extracted, pillars drawn, and the panels closed and 
sealed off independently of each other. Greater security is 
afforded against creep and squeeze. Coal that disintegrates 
on standing can be quickly worked out. 

Bearing In, or Undercutting.—In any method of mining 
where the coal is undermined, advantage should be taken of 
the roof pressure to assist in breaking down the coal and in 
bearing in. The fact is often overlooked that the roof pressure 
upon the face coal makes it brittle and more susceptible to the 
pick, and the good miner starts a shallow mining in the under 
clay, or lower coal, and carries it the entire width of the face. 
Such a gradual system of mining throws the pressure on the 
coal face gradually, and the coal breaks in larger pieces. The 
depth of the undercut depends on the thickness of the seam and 
the other conditions. 

SYSTEMS OF WORKING COAL 

There are two general systems of working coal seams, the 
room-and-pillar, and the longwall. There are, however, a 
great number of modifications of each so that it is often difficult 
to exactly classify a given method. 

Room-and-Pillar System.—The room-and-pillar system, also 
known as the pillar-and-chamber or bord-and-pillar , is the oldest 
system, and the one very generally used in the mines of the 
United States. The coal is first mined from a number of com¬ 
paratively small places called rooms, chambers, stalls, bords, etc., 
which are driven either square from or at an angle to the haul¬ 
ageway. These openings may be wide or narrow, and may be 
a roadway, incline, or chute, according to existing conditions. 
The pillars that are left between the openings in the original 
workings support the roof, and usually are subsequently 
removed. All forms of room-and-pillar workings become 
impracticable when the thickness of the pillars necessary to 
support the roof pressure much exceeds double the width of the 
breast openings. 


MINING 


203 


Fig. 1 shows a typical room-and-pillar method for working 
an approximately horizontal seam of coal of moderate thickness 
(4 to 10 ft.), and with a fairly good roof and bottom. The 
room openings are made suitable to prevailing conditions. 
The width of the room and the form of the opening depend on 
the character of the roof and the extent to which it is necessary 
to leave a pillar to support the cross-heading, it being advan¬ 
tageous, of course, to open out the room to its full width at the 
earliest possible moment. 


The typical room-and-pillar plan, Fig. 1, shows the main 
headings and the rooms driven parallel to the direction of the 
dip, and the cross-headings parallel to the strike, but in most 
coal seams there are vertical cleavages, called cleats, which 
cross the coal in two directions about at right angles to each 
other. Face cleats are the more pronounced, while the end, or 
butt, cleats are the shorter, less pronounced joints. The direc¬ 
tion of the face with respect to the cleats is of prime importance 
as greatly facilitating, or retarding the mining of the coal. 



204 


MINING 



Fig. 2 


Fig. 2 shows the different positions that the face may occupy 
with respect to the direction of the cleats. The angle of the 
breast depends on the hardness of the coal and freedom of the 
cleats, and each method has its peculiar adaptation to the vary¬ 
ing conditions of the coal seam. When the face cleats are work¬ 
ing free and the coal is very soft, it may be necessary to drive 

end on. The end-on 
method is best adapted 
to a very heavy roof 
pressure, while for a 
light roof pressure the 
short-horn method as¬ 
sists in breaking the 
coal. If the face cleats 
are free and the coal 
breaks readily along 
them, and it is reason¬ 
ably hard, the long-horn method is adopted, for when the coal is 
undercut it needs more support than it gets from the cleats, and 
its weight must be thrown somewhat upon the end cleats. 
Face on is adopted when the face cleats are not as free or numer¬ 
ous as the butt cleats. 

The short-horn method is adapted to heavy roof pressure and 
wide room pillars, as the face cleats are here quite pronounced, 
and the pillars between the rooms thereby weakened to a large 
extent; hence, wide pillars are more often employed when 
working on the ends of the coal. 

Longwall Method of Mining.—In the longwall system, no 
portion of the seam is allowed to remain after leaving the vicin¬ 
ity of the shaft. The method depends on producing a uniform 
and gradual settlement of the roof a few yards behind the work¬ 
ing face. In starting, the work of extraction may begin at the 
shaft itself, the coal being taken out all around and its place 
filled with solid packs, leaving only space for the roadways; or 
a pillar of solid coal cut only by the roadw'ays, may be left to 
support the shaft. The longwall work may then be started 
uniformly all around this pillar. 

Longwall may be advancing or retreating. In longwall 
advancing, mining begins at or near the foot of the shaft and 




MINING 


205 


advances outwards, forming a gradually widening and increas¬ 
ing length of face to the boundary. The passages are made 
through the excavated portions of the mine, and are maintained 
by pack walls built either of the refuse secured in mining or 
from material brought in from the surface. Pack walls are 
built on each side of the roadways, and at regular intervals in 
the gob or waste area, and the roof settles firmly on these packs, 
pressing them into the bottom, or compressing them until the 
roof subsidence is complete. The height of the main roadway 
is maintained by brushing the roof or lifting the bottom. Long- 
wall advancing is better suited to thin seams than to thick ones, 
to flat rather than pitching, and to good roofs and hard floors. 
In longwall retreating, entries, gangways, or headings are driven 
to the boundary or to other convenient distances inbye, and 
the pillars between these entries are then drawn back toward 
the shaft; this is also called working home. Longwall retreating 
is adapted to thick beds; to those liable to gob fires; to seams 
of hard coal having a considerable pitch; to pockety, or irregu¬ 
lar seams; and to a soft and treacherous top. The air-course 
is also less broken along the face, and better haulage installa¬ 
tions can be made. Its chief disadvantage is the large amount 
of dead work necessitated before returns are received. There is 
no expense in keeping up the haulage road so far as creep or 
falling roof is concerned, as the roads are all in solid coal, nor 
is there any trouble from gob fires or water; and little detri¬ 
ment to the working face is caused by the mine having to stand 
idle for a time. If the seam is high enough for the mules or 
horses, no rock whatever will need to be taken down. The 
coal seam will be proved before 10% of it is extracted. The 
ventilation in the retreating plan is as near perfect as it is pos¬ 
sible to get it in practice. All the airways are tight, a thing 
impossible to get in the advancing plan; and it is a compara¬ 
tively easy matter to shut off fire or to allow a portion of the 
working face to remain idle. 

Longwall retreating is frequently used for working quite 
limited sections of a mine in which the seam of coal is 16 to 20 
ft. thick, and, inclined not more than 10°. A series of 8 or 10 
pairs of headings are turned off the butt entry and driven a 
distance, dependent on local conditions, where the working face 


206 


MINING 


is formed by driving cross-cuts from one to the other. This 
face is carried back on the retreating plan, allowing the roof 
to cave in or settle on the gob as the work approaches the butt 
entry. In this way, any extra weight, that would crush and 
ruin the adjacent coal is avoided. This method is also used in 



wT' 

JBr 

I 


a 1 


v‘/r> K nV^.T/Vm 'G<rf^&Ky.\7\s <! 


Fig. 3 


lower seams in which the coal is soft, or the roof, or bottom, or 
both, are of such a nature as to give trouble in working the room- 
and-pillar system. Sometimes, instead of driving pairs of 
headings at considerable distances apart, a number of single 
headings are driven comparatively close together, and connected 



MINING 


207 


by cross-cuts from 10 to 20 yd. apart. When the limit of the 
section is reached, the working face is formed and carried back, 
as in the other plan. This latter method is more suitable for 
tender roof, or a coal in which the face and butt cleats are not 
prominent. 

Fig. 3 shows a plan of combined longwall advancing and 
retreating. In the upper arrangement, or Scotch longwall, 
the face is semicircular and the roads are turned off at angles 
of 45°. This plan is suitable for seams up to 3 ft. thick with 
a weak top, which pitch less than 20° and is situated at almost 
any depth. It is the one from which most of the longwall prac¬ 
tice in the central coal basins of the United States is taken. 
In the lower arrangement headings from 200 to 300 ft. apart are 
driven in pairs to the boundary. Such a combination of long¬ 
wall advancing and retreating insures qp unvarying supply of 
coal, for while one side continually leaves the shaft, the other 
approaches it. 

Timbering a Longwall Face.—The method of timbering the 
working face depends on the nature of the roof, floor, coal, 
etc. The action of the roof on the coal face is regulated almost 
entirely by timber; consequently, when the coal is of such a 
* nature as to require little weight to make it mine easily, the 
roof must be timbered with rows of chocks and, if necessary, a 
few props. 

Cribs, Pack Walls, and Stowings.—Pack walls should be 
built large enough at first and kept well up to the face, to pre¬ 
vent the weight coming upon the timber and also to permit the 
roof to settle rapidly when the timber is taken out of the face. 
Often the roof will not stand this second movement without 
breaking, and possibly closing in the entire face. The face 
should therefore be kept in shape, and just as soon as there is 
room for a prop or chock, it should be put in immediately, and 
the pack walls likewise should be extended after each cut or 
web is loaded out. No waste material, except such as will 
hasten spontaneous combustion, should be taken out of the 
mine to the surface. 

Control of Roof Pressure.—The working face of a longwall 
working should advance up grade, but this face cannot always 
be kept parallel with the strike. When the angle at the line of 


208 


MINING 


face, made with the line of strike, is less than 90°, the greater 
pressure of the covering rocks is thrown on the gob; when this 
angle is more than 90°, the greater pressure comes on the coal. 
The angle made by the working face with the line of pitch varies 
inversely as the vertical angle of pitch, or for a high pitch this 
angle is small and for a low pitch it is large. Where longwall is 
worked in adjacent sections, care must be taken to prevent the 

advancing of one section throwing 
a crushing weight on any of the 
others, and thus producing a crush 
or an uncontrollable cave. Where 
the rocks are pitching, and a greater 
portion of the cracks that cut them 
run in lines parallel to the strike, 
neither stone nor timber can effi¬ 
ciently support the roof, which fre¬ 
quently breaks off close to the working face. 

The ends of all stone packs nearest the face of the coal should 
be in line, and the ends of these pack walls should form a line 
parallel to the face of the coal. Timbers set at equal distances 
and in line along a longwall face are much more efficient in 
supporting the roof than irregularly set timbers. Fig. 4 shows 
the proper way of locating the pack walls and the face timber. 

NUMBER OF ENTRIES 

The entries in a mine nay be driven single, double, triple, etc. 
The single-entry system is only advisable under certain condi¬ 
tions and for short distances because the ventilation must be 
maintained along the face of the rooms, and there is but one 
haulage way, which may easily be closed by a fall or creep. 
Rooms are turned off on one or both sides of the entry. The 
double-entry system is most commonly used. Two parallel 
entries are driven, separated by an entry pillar whose thickness 
varies with the depth of the seam, and connected at intervals 
of about 20 yd. by cross-cuts or breakthroughs to maintain 
ventilation. The triple-entry system is used particularly in very 
gaseous seams requiring separate return airways; or, at times, 
in mines where the large output requires ample haulage roads. 
It is usually applied to the main entries only, but sometimes, 











MINING 


209 


also, to the cross-entries. In gaseous mines, the middle entry 
is usually made the haulage road and intake airway, and the 
outside entries the return air-courses for either side of the mine, 
respectively. A still larger number of entries even has been 
suggested for deep workings where it is difficult to keep open 
broad passages, but these have not been generally adopted or 
tried experimentally to any great extent. 


PILLARS 

It is impossible to give exact rules or formulas for determining 
the proper size of pillars. Each case in practice requires special 
consideration, and in laying out the pillars in a virgin field it is 
well to find out w'hat the current practice is in similar fields. 
In general, the thicker the seam and the greater its depth from 
the surface, the greater should be the thickness of the pillar. 

Shaft Pillars.—Various formulas have been given to deter¬ 
mine the size of shaft pillars, and the results given by these 
several formulas are very diverse. 


Merivale's Formula .— 



in which 5 = length of side of pillar, in yards; D depth of shaft, 
in fathoms. 

Andre's Formula. —Up to 150 yd. depth, have the pillar 35 
yd. square, and for greater depths increase 5 yd. on each side 
for every 25 yd. of increased depth. 

Dron’s Formula. —Draw lines enclosing all surface buildings 
that it is necessary to erect about the head of the shaft, and 
make the shaft pillar so that solid coal will be left outside these 
lines all around for a distance equal to one-third the depth of 
the shaft. 

Wardle’s Formula. —Shaft pillars should not be less than 40 
yd. square down to a depth of 60 fath. and should increase 10 
yd. on a side for every 20 fath. increase in depth. 

Hughes' Formula. —Leave 1 yd, in width of pillar for every 
yard in depth of shaft. 

Pamely’s Formula. —Allow a pillar 40 yd. square for any 
depth up to 100 yd.; for greater depths, increase the pillar 
5 yd. for every 20 yd. in depth. 


210 


MINING 


Calculating the size of pillar from each of these authorities 
gives the results shown in the accompanying table. 


SIZE OF SHAFT PILLARS 


Authority 

For Shaft 

300 Ft. Deep 

For Shaft 

600 Ft. Deep 

Merivale. 

22 yd. square 

31 yd. square 

Andre. 

35 yd. square 

45 yd. square 

Wardle. 

40 yd. square 

60 yd. square 

Pamely. 

40 yd. square 

65 yd. square 

Dron. 

331 yd. square* 

66§ yd. square* 

Hughes. 

100 yd. diameter 

200 yd. diameter 


*Outside of buildings. 


None of these formulas takes account of the thickness of the 
seam, and the following formula, which takes account of this 
very important element, was suggested by Mr. R. J. Foster, 
in Mines and Minerals: 

Radius of pillar = 3 V DXt , 
in which D = depth of shaft; 

t = thickness of seam. 

Pitching seams require smaller pillars on the low side than 
on the rising side of the shaft. 

Room Pillars. —The relative width of pillar and breast is 
dependent on the weight of cover, as compared with the char¬ 
acter of the roof and floor, and the crushing strength of the 
coal. These relative widths are determined largely by prac¬ 
tice. Speaking generally, the narrower the rooms or chambers, 
the higher is the cost in yardage, the greater the production of 
slack and nut coal, the greater the consumption of powder, 
track iron, ties, etc., and the greater the cost of dead work. 

For bituminous coal of medium hardness and good roof and 
floor, a rule often used is to make the thickness of room pillars 
equal to 1% of the depth of cover for each foot of thickness of 
the seam, according to the expression 

t 















MINING 


211 


in which Wp = pillar width; 

/ = thickness of seam; 

D = depth of cover. 

Then the width of breast or opening is made equal to the 
depth of cover divided by the width of pillar thus found, 
according to the expression 



in which W 0 = width of room. 

The accompanying table is for first working, with the design 
of afterwards taking out the pillars, the width of the principal 
workings being 5 yd., and cross-holings 2 yd. 


DUNNS’ TABLES OF SIZE OF ROOM PILLARS FOR 
VARIOUS DEPTHS 


Depth 

Feet 

Size of 
Pillars 

Yards 

Pro¬ 

portion 

in 

Pillars 

Depth 

Feet 

Size of 
Pillars 

Yards 

Pro¬ 

portion 

in 

Pillars 

120 

20 X 5 

.41 

1,080 

26X14 

.69 

240 

20 X 6 

.50 

1,200 

26X16 

.71 

360 

22 X 7 

.52 

1,320 

28X18 

.73 

480 

22 X 8 

.57 

1,440 

28X20 

.75 

600 

22 X 9 

.59 

1,560 

30X21 

.77 

720 

22X12 

.61 

1,680 

30X22* 

.78 

840 

26X15 

.63 

1,800 

30X24 

.79 

960 

28X16 

.66 





Extremely large pillars must often be left as a precautionary 
measure to protect permanent haulage ways and surface 
buildings, or to avoid any possibility of a break in the roof that 
would cause an inflow of water. 

Drawing Pillars.—Drawing pillars is about the most danger¬ 
ous work the miner has to perform, but the fact of its being 
so is no doubt the reason why, comparatively speaking, so 
few serious accidents happen in it. It is not so much that the 
best, most skilled workmen are chosen to perform pillar drawing, 
as that the men, being alive to the dangers, are more on the 
alert and careful to protect themselves. Methods of drawing 














MINING 


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MINING 


213 


pillars vary according to the inclinations of the seams, the 
nature of the roof and floor, and the character of the coal; 
Figs. 1 and 2 show the common methods. In Fig. 1, at A, 
B, and C, the drawing begins by cross-cutting the fast ends 



of the pillars to obtain a retreating face. At A is shown a 
method for soft coal and narrowing pillars; at B, a method for 
wide pillars, the end being taken in two lifts; while the method 
at C is for harder coal and shows it taken in three lifts. At 
D and E the pillars are cut into stocks to be drawn by side or 
end lifts, according to the character of the coal, the inclination 



Fig. 2 


of the seam, thickness of the cover, and the strength or weak¬ 
ness of the roof and floor. 

Fig. 2 shows some of the methods used in robbing the pillars 
in steep pitching, thick beds of anthracite. To get the coal 














214 


MINING 


out of the pillar at the left of A, a skip is taken off the side, as 
shown. Successive skips are thus taken off until the whole is 
removed, the miner keeping the manway open to the heading 
below as a means of retreat. The pillar between A and B 
is similarly worked. To remove that between B and C, a nar¬ 
row chute or heading is driven up the middle, and cross-cuts 
put to the right and left a few yards from the upper end. 
Shots are placed in the four blocks of coal thus formed, as 
shown, and they are fired simultaneously by battery. This 
operation is repeated in each descending portion unless the 
pillar begins to run. A pillar from which the coal has started 
to run is shown to the right of C. 

SPONTANEOUS COMBUSTION 

The cause of the spontaneous ignition of coal, chiefly, is the 
condensation and absorption of oxygen from the air by the 
coal, which of itself causes heating, and this promotes the 
chemical combination of the volatile hydrocarbons in the coal 
and some of the carbon itself with the condensed oxygen. 
Another cause is moisture acting on sulphur in the form of 
iron pyrites. The heating effect of this cause is very small, 
and it acts rather by breaking the coal and presenting fresh 
surfaces for the absorption of oxygen. 

Gob fires are due to the spontaneous ignition of coal, and 
are most likely to occur in pack walls and gobs where there 
is an insufficiency of air. Ample ventilation is the best 
preventive. 

COAL STORAGE 

The coal store should be well roofed in, and have an iron 
floor bedded in cement. All supports passing through and in 
contact with the coal should be of iron or brick; if hollow iron 
supports are used, they should be cast solid with cement. 
The coal must never be loaded or stored during wet weather, 
and the depth of coal in the store should not exceed 8 ft.. and 
should only be 6 ft. where possible. Under no condition must 
a steam or exhaust pipe or flue be allowed in or near any wall 
of the store, nor must the store be within 20 ft. of any boiler, 
furnace, or bench of retorts. No coal should be stored or 
shipped to distant ports until at least 1 mo. has elapsed since 


MINING 


215 


it was brought to the surface. Every care should be taken 
during loading or storing to prevent breaking or crushing of 
the coal, and on no account must a large accumulation of small 
coal be allowed. These precautions, if properly carried out, 
would amply suffice to entirely do away with spontaneous 
ignition in stored coal on land. 

When the coal pile has ignited, the best way to extinguish 
the fire is to remove the coal, spread it out, and then use water 
on the burned part. The incandescent portion is invariably 
in the interior, and when the fire has gained any headway 
usually forms a crust that effectually prevents the water from 
acting efficiently. 

MODIFICATIONS OF ROOM-AND-PILLAR METHODS 

Some modifications of the room-and-pillar plan shown in 
Fig. 1 can usually be applied to seams whose dip does not exceed 
3°. When the pitch is greater, rooms are often turned off 
toward the rise only, and the cross-entries driven correspond¬ 
ingly closer together. When the pitch is from 5° to 10°, 
the cars may still be taken to the face if the rooms are driven 
across the pitch, thus making an oblique angle with an entry 
or gangway, the rooms being known as room, breasts. 

Buggy Breasts. —For inclinations between 10° and 18°, that 
is, after mule haulage becomes impossible and until the coal 
will slide in chutes, buggies are often used. Fig. 1 shows a 
buggy breast, in plan and section. Coal is loaded into a small 
car or buggy c, which runs to the lower end of the breast and 
there delivers the coal upon a platform /, from which it is loaded 
into the mine car. The refuse from the seam is used in building 
up the track, and if there is not sufficient refuse for this, a 
timber trestle is used. 

Another form of buggy breast is shown in Fig. 2. Here 
the coal is dumped directly into the mine car from the buggy. 
If the breast pitches less than 6°, the buggy can be pushed to 
the face by hand, but in rooms of a greater pitch, a windlass 
is permanently fastened to timbers at the bottom of the breast, 
while the pulleys at the face are temporarily attached to the 
props by chains, so that they can be advanced as the face 
advances. The rope used is from % to f in. in diameter, and. 


216 


MINING 


any form of ordinary horizontal windlass can be used. With 
the windlass properly geared, one man can easily haul a 
buggy to the face of a breast in a few minutes. The buggy 
runs upon 20-lb. T rails spiked with 2\ in.Xf in. spikes upon 
2 in.X4 in. hemlock studding sawed into lengths of 14 ft. 

Chute Breasts. —Seams pitching more than 15° are usually 
worked by chutes, or self-acting inclines. When the pitch is 
between 15° and 30°, sheet iron is laid to furnish a good slid¬ 



ing surface for the coal. On inclinations of less than 18° 
to 20°, it is usually necessary to push the coal down the chute. 
Sheet iron is not required on pitches above 30°. It must be 
remembered that these pitches are only fair averages, as much 
depends on the character of the coal. Anthracite slides more 
easily than bituminous. 

To secure the best returns from a coal seam, the slope or 
shaft should be driven to the basin, and the lowest gangways 
or levels first driven to the property limits, and the coal then 






























MINING 


217 


worked retreating toward the slope or shaft. Practice is, 
however, usually contrary to this, and the upper levels or 
gangways are turned off first, and working places opened out 
as rapidly as the. gangway is driven. Fig. 3 shows a method 
of grouping rooms that may be used where the pitch is from 
8° to 20°, the straight heading being driven on the strike 



Fig. 2 

and the other headings at such angles as will give a good grade 
for haulage purposes. 

Pillar-and-Stall Method. —The pillar-and-stall system is a 
modification of the room-and-pillar, to which it is similar in 
all respects excepting in the relative size of the pillars and 
breasts. The stalls are usually opened narrow and widened 





























218 


MINING 


inside, according to conditions of roof, floor, coal, depth, 
etc., being from 4 to 6 yd, in the single-stall method, with the 

pillars about the same 
width. Fig. 4, A and B, 
shows single and double 
stalls. This system is 
adapted to weak roof 
and floor, or strong roof 
and soft bottom, to a 
fragile coal, or wherever 
ample support is re¬ 
quired, and is particu¬ 
larly useful in deep 
seams with great roof 
pressure. Double stalls 
are often driven from 12 to 15 yd. wide, with an intervening 
pillar of sometimes 30 yd. Following are a few applications 
of the pillar-and-room method carried out in some of the leading 
coal fields of America: 

Connellsville Region.—Fig. 5 shows the common method used 
in Connellsville, Pa., region, where the average dip is about 5%. 
The face and butt headings are driven, respectively, at right 
angles to each other on 
the face and the butt 
of the coal. The face 
headings leave the 
main butts about 
1,000 ft. apart, while 
from these face head¬ 
ings, and 400 ft. apart, 
secondary butts are 
■driven, and again from 
these butts on the face 
of the coal the rooms 
or wide workings are 
excavated to a length 
of 300 ft., this having 
proved the most convenient length for economical working. 
Room pillars have a thickness of 30 to 40 ft., while the rooms 



Fig. 4 



Fig. 3 








MINING 


219 


are 12 ft. in width and are spaced 42 to 52 ft. between centers, 
depending on depth of strata over the coal. The headings are 
8 ft. wide, and in all main butts and faces the distance between 
centers of parallel headings is 60 ft., leaving a solid rib of 52 ft. 



Fig. 5 


A solid rib of 60 ft. is also left on the side of each main 
heading. 

The method of drawing ribs is one of the advantages of the 
system, as it is harder to do successfully in a soft coal like the 
Connellsville coal than in hard coal. The coal itself is firm. 



220 


MINING 


When necessary to protect the top or bottom, 4 to 6 in. of 
coal are left covering the soft material. 

The method just given is often applied to a whole series of 
butts (4 or 5) at once instead of to each butt in turn. In this 
case, work is started at the upper end of the uppermost butt 
and progresses, as shown; but, after cutting across the butt 
heading from which the rooms were driven, the butt head¬ 
ing itself and the upper rooms from the second butt, or that 
just before, are drawn back by removing continuous slices 
from the rooms of the upper butt, and on across the next 
lower butt, etc., all on an angle to the butts, until another 
butt is reached, etc. This gradually makes a longer line of 
fracture, which is only limited by the number of butts it is 
desired to include at one time in the section thus mined. 

Pittsburg Region.—The coal is worked in much the same way 
as in the Connellsville region, except that a different system 
of drawing ribs is used. The coal is worked on the room-and- 
pillar system, with double entries, with cut-troughs between 
for air, and on face and butt. Entries are about 9 ft. wide, 
and the rooms 21 ft. wide and about 250 ft. long; narrow (or 
neck) part of room, 21 ft. long by 9 ft. wide. Room pillars 
are 15 to 20 ft. wide, depending on depth of strata over the coal, 
which is from a few feet to several hundred feet. The mining 
is done largely by machines of various types. Coal is hard, 
of course, and, in many places, the roof immediately over the 
coal is also quite hard. There are about 4 ft. of alternate 
layers of hard slate and coal above the coal seam. Rooms 
are mined from lower end of butt as fast as butt is driven, 
the ribs being drawn as mining progresses. As the coal is 
harder than in the Connellsville region, thickness of coal pillar 
between parallel entries is somewhat less. 

Clearfield Region.—The butt and face are not strongly 
marked in the B or Miller seam, the one chiefly worked in the 
Clearfield region. Where possible, these cleavages are followed 
in laying out the workings, but the rule is to drive to the great¬ 
est rise or dip and run headings at right angles to the right 
and left, regardless of anything else. The main dip or rise 
heading is usually driven straight, and is raised out of swamps 
or cut down through rolls—very common here—unless they 


MINING 


221 


are too pronounced, when the heading is curved around them. 
The same is true of room headings, except that they are more 
usually crooked, not being graded except over very minor 
disturbances. 

Reynoldsville Region. —The average thickness of the prin¬ 
cipal seam is 6| ft. and the pitch is 3° to 4°. The coal is hard 
and firm, and contains no gas; the cover is light, and on the 
top of the coal there are 3 or 4 ft. of bony coal; the bottom is 
fireclay. Drift openings and the double-entry system are 
used. Both main and cross entries are 10 ft. wide, with a 
24-ft. pillar between. The cross-entries are 600 ft. apart, and 
a 24 ft. chain pillar is left along the main headings. The 
rooms are about 24 ft. wide and open inbye, the necks being 
9 ft. wide and 18 ft. long. The pillars are from 18 to 30 ft. 
thick. 

West Virginia. —In the northern part of West Virginia, the 
coal measures vary from 7 to 8 ft. in thickness, and have a 
covering varying from 50 to 500 ft. The coal does not dip at 
any place over 5%. In most places the coal is practically level, 
or has just sufficient dip to afford drainage. The usual method 
of exploitation is to advance two parallel headings, 30 ft. apart, 
on the face of the coal. At intervals of 500 to 600 ft., cross¬ 
headings are turned to right and left, and from these headings 
rooms are turned off. These cross-headings are driven in pairs 
about 20 or 30 ft. apart. Between the main headings and the 
first room is left a block of coal about 100 ft., and on the cross¬ 
headings there is often left a barrier pillar of 100 ft. after every 
tenth room. 

The headings are driven from 8 to 12 ft. wide, and the rooms 
are made 24 ft. wide and 250 to 300 ft. long. A pillar is left 
between the rooms about 15 to 20 ft. wide. These pillars are 
withdrawn as soon 1 as the panel of rooms has been finished. 
The rooms are driven in from the entry about 10 ft. wide for a 
distance of 20 ft., and then the width is increased on one side. 
The track usually follows near the rib of the room. Cross-cuts 
on the main and cross headings are made every 75 to 100 ft., 
and in rooms about every 100 ft. for ventilation. 

The double heading system of mining and ventilation is in 
vogue. Overcasts are largely used, but a great many doors are 


222 


MINING 


used in some of the mines. Rooms are worked in both direc¬ 
tions, when the grades are slight, but when the coal dips over 
1%, the rooms are driven in one direction only; in this case, 
they are made as much as 350 ft. long. It is the custom then 
to break about every third room into the cross-heading above. 
The floor of this bed of coal, being composed of shale and fire¬ 
clay, often heaves, especially when it is made wet. Some 



Fig. 6 


trouble is at times experienced by having the floor heave by 
reason of the pillars being too small for the weight they support. 

Alabama Methods. —Fig. 6 shows the common methods used 
in working the Alabama coals. The seams now working vary 
from 2 to 6 ft. thick, and they pitch from 2° to 40°. Where the 
seams are thin, the coal is hard, and pillars of about 20 to 30 ft. 
are used to support the roof. The rooms are worked across the 
pitch on an angle of about 5° on the rail, Fig. 6, A, when the 



MINING 


223 


coal does not pitch greater than 20°; where the pitch is greater, 
chutes are worked and the rooms are driven straight up the 
pitch, as in Fig. 6, B. In a few cases, where the pitch is not 
greater than 15°, double rooms are worked with two roadways 
in each room, as in Fig. 6, C. 

George’s Creek District, Md, —Fig. 7 shows the method used 
in the George’s Creek field, Md. The coal shows no indication 
of cleats, and the butts and headings can be driven in any 
direction. The main heading is driven to secure a light grade 
for hauling toward the mouth. Cross-headings making an 
angle of 35° to 40° are usually driven directly to the rise, and of 
dimensions shown. 


nSKSk 



Fig. 7 


Indiana Coal Mining. —Fig. 8 shows the method as used 
in Indiana. The entries are generally 6 ft. high, 8 ft. broad, 
the minimum height required by law being 4 ft. 6 in. The 
rooms are from 21 to 40 ft. in width. The mines are generally 
shallow. The rooms in Fig. 8 are shown as widened on both 
ribs, but a more usual method in this locality is to widen the 
room on the inbye rib, leaving one straight rib for the protection 
of the road in the room. 

Iowa Coal Mining. —The entry pillars along the main roads 
are 6 to 8 yd. thick, for the cross entries 5 to 6 yd., and for the 
rooms 3 to 5 yd. Room pillars are drawn in when approaching 
a cross-cut. Both room-and-pillar and longwall methods are 
in use, with modifications of each. In the room-and-pillar 
method, the double-entry system is almost invariably used in 





224 


MINING 


the larger mines. Rooms are driven off each entry of each pair 
of cross-entries at distances of 30 to 40 ft., center to center. The 
rooms are 8 to 10 yd. in width, and pillars 3 to 4 yd. The 
rooms are narrow for a distance of 3 yd., and then widened inbye 
at an angle of 45° to their full width. They vary from 50 to 
100 yd. in length, and the road is carried along the straight rib. 

When double rooms are driven, the mouths of the rooms are 
40 to 50 ft. apart, and they are driven narrow from the entry 
a distance of 4 or 5 yd. A cross-cut is then made connecting 
them, and a breast 16 yd. wide is driven up 50 to 60 yd. The 



MAIN ENTRY 


ROOM 


f 6*fAK THROUGH 


RAILS 


FACE Vv 


Fig. 8 

pillar between each pair of rooms is 12 to 15 yd. In some cases, 
the stalls are usually turned off narrow and widened inside, the 
pillar varying from 5 to 8 yd. The stalls are 30 to 40 yd. in 
length, and the pillars are drawn back. When the stalls are 
driven in pairs, the pillar 8 to 10 yd. in width is carried between 
them. 

When the longwall system is used, the main haulage road 
runs in each direction from the foot of the shaft, and on both 
sides of this diagonal roads are turned at an angle of 45°, or 
parallel to the main haulageway. These are spaced 10 yd. 
apart and driven 50 to 60 yd., when they are cut off by another 































MINING 225 

diagonal road. Panel breasts are used where the conditions 
are such as to induce a squeeze. Rooms are turned narrow 
off entries and are arranged in sets of 6 to 12 rooms, with 
a pillar 10 to 20 yd. wide between the sets of rooms. When 
the rooms have progressed a short distance from the entry, 
they are connected by cross-cuts, and the longwall face is 
carried forwards from this point. Packs are built and the roof 
allowed to settle, as in longwall. The wide pillars are taken 
out after the roof has settled. 



Manway fc/f/r Course 
> r~ a,./-.. 


Chafe 




Fig. 9 

Tesla, Cal. —The Tesla, Cal., method is shown in Fig. 9. 
The coal seam averages 7 ft. of clear coal, and pitches 60°. 
This system was adopted in a portion of the mine to get coal 
rapidly; for, at this point, a short-grained, slate cap rock came 
in over the coal, making it difficult to keep props in place. The 
floor is a close blue slate and has a decided heaving tendency. 
The roof is an excellent sandstone. There is a small but 
troublesome amount of gas. Two double chutes are driven up 
the pitch at a distance of 36 ft. apart, connected every 40 ft. by 















226 


MINING 


cross-cuts. One side of each chute is used for a coal chute and 
the other for a manway and air-course. At a distance of 12 yd. 
apart small gangways are driven parallel with the main mine 
gangways. These are continued from each chute a distance 
of 300 ft., if the conditions warrant it. The top line is then 
attacked from the back end and the coal is worked on the 
cleavage planes; the breast, or room, therefore consists of a 


/V? a Seam 



Fig. 10 


12-yd. face, including the drift or gangway through which the 
coal is carried to the chutes; a rib of coal (2 or 3 ft.) is left 
between the breasts to keep the rock from falling on the breast 
below. Thus in each breast the miners have a working face of 
about 15 or 16 yd., and as the coal is directed to the car by a 
light chute, moved along as the face advances, the coal is 
delivered into the cars at small cost, and but little loss results 




MINING 


227 


from the falling coal, as a minimum of handling is thus obtained. 
Fig. 10 shows another system used in No. 7 vein at the same 
place. The seam averages 7 ft. of coal. The roof is shelly and 
breaks quickly, hence the coal must be mined rapidly. 



B C Z> 

Fig. 11 


New Castle, Colo. —The following method is used at New 
Castle, Colo., for highly inclined bituminous seams. The coals 
mined are only fairly hard, contain considerable gas, and make 
much waste in mining. Fig. 11 shows the method used for 
extracting the Wheeler or thicker vein to its full width of 45 ft., 



















228 


MINING 





Fig. 12 






































MINING 


229 


and the E seam 18 ft. thick, excepting that left for pillars. 
Rooms and pillars are laid out under each other in the two seams 
whenever practicable. Entries are along the foot-wall; 30 ft. 
up the pitch is an air-course. Rooms and breasts are laid out 
as shown in B and C. 

MODIFICATIONS OF LONGWALL METHOD 

Fig. 12 shows a good arrangement of the main and temporary 
haulageways in a flat seam. The chief object in any plan of 
longwall workings is to have the permanent roadways the 
arteries of the system, providing the most direct route from all 
sections of the mine to the shaft. The temporary roads or 
working places are only maintained for a distance of 60 to 100 
yd., until cut off by subroads branching at regular intervals from 
the main roads. The full heavy lines indicate the permanent 
haulageways, except only the main intake airway (12 ft. wide), 
running west from the downcast shaft D, and the main return 
air-course (12 ft. wide) leading from the face on the east side 
to the manway around the upcast U, which is the hoisting shaft. 
The full light lines indicate the diagonal subroads, driven to 
cut off the working places, shown by the dotted lines. The 
stables are located as shown in the shaft pillar, between the two 
shafts, where they will not contaminate the air going into the 
mine, but will receive air fresh from the downcast and discharge 
it at once into the upcast current. This position also affords 
ready access from either shaft in case of accident, and for the 
handling of feed and refuse. The pumps may be located in any 
convenient position at the foot of the upcast. The shaft 
bottoms are driven 14 ft. wide nearly through the shaft pillar, 
and are continued 10 ft. wide north and south through the gob. 
The width of all other roads and subroads is made 8 ft. 

METHODS OF MINING ANTHRACITE 

A perfectly flat seam of anthracite is seldom found in America, 
and even where a portion of the seam may be found lying com¬ 
paratively flat, such sudden changes in dip must be expected 
that a system adapted to working on a pitch is almost univer¬ 
sally used. A breast may start on a low pitch and the pitch 
may increase gradually until it becomes vertical, or the reverse 


230 


MINING 


may be the case. The cleat is usually lacking in anthracite, 
and the direction of driving the breasts is determined largely 
by the pitch and by haulage considerations. 

For pitches up to 30°, the methods already described are, 
in general, applicable, with certain changes due to local con¬ 
siderations. There is considerable difference in the methods of 
opening rooms in anthracite and bituminous mines, owing to 
variations in the characteristics of the coals and to the fact that 
anthracite will slide on chutes of less inclination than bituminous 
coal. Where the pitch does not exceed 4°, the rooms are turned 
off at right angles to the gangway. In moderately thick coal 
seams, pitching between 4° and 18°, the rooms are generally 
driven across the pitch, forming room breasts, thus securing a 
grade that permits the haulage of the cars to the face. 

There are two methods of mining thick coal in breasts when 
nearly flat: (1) The breasts are opened out and driven to the 
limit in the lower bench of coal, and the top benches are blown 
down afterwards, beginning at the face and working back. 
(2) When the roof is good and there is no danger of its falling 
and closing up the workings, the upper benches may be worked 
in the opposite direction, beginning at the gangway and driving 
toward the limit of the lift, or the working of the upper bench 
may follow up that of the lower bench. When the seam is less 
than 12 ft., the top is supported by props; in thicker seams, the 
expense is so great for propping that but little attempt is made 
to support the roof. In the thicker anthracite seams (notably 
the Mammoth), the coal in the breasts is so worked as to make 
an arch of the upper benches of coal, which acts as a temporary 
support for the roof, the coal in the arch being extracted when 
the pillars are robbed. 

When the inclination of anthracite seams is less than 30°, the 
breasts may be opened with one chute in the center, which ends 
in a platform projecting into the gangway, off which the coal 
can be readily loaded into the mine car. When this method is 
employed, the refuse is thrown to either side of the chute. If 
the pillars are to be robbed by skipping or slabbing one rib 
only, most of the refuse is kept on one side. Sometimes, when 
the top is good, and the breasts are driven wide, two chutes are 
used, but the cost of making the second chute is considerable 



MINING 231 

and is therefore not advisable unless necessitated by the method 
of ventilation employed. 

Fig. 1 shows a panel system that gives good results in thick 
seams pitching from 15° to 45°, where the top is brittle, the coal 
free, and the mine gaseous. Rooms or breasts are turned off 
the gangway in pairs, at intervals of about 60 yd. The breasts 
are about 8 yd. wide, and the pillar between about 5 yd. wide, 
which is drawn back as soon as the breasts reach the airway, 
near the level above. In the middle of each large pillar between 
the several pairs of breasts, chutes about 4 yd. wide are driven 
from the gangway up to the airway above. These are provided 
with a traveling way on one side, giving the miners free access 



Fig. 1 

to the workings. Small headings are driven in the bottom 
bench of coal, at right angles to these chutes, and about 10 or 
20 yd. apart. These headings are continued on either side of 
the chutes until they intersect the breasts. When the chute 
and headings are finished, the work of getting the coal in the 
panel is begun by going to the end of the uppermost heading and 
widening it out on the rise side until the airway above is reached 
and a working face oblique to the heading is formed. This face 
is then drawn back to the chute in the middle of the panel. 
After the working face in the uppermost section has been drawn 
back some 10 or 12 yd., work in the next section below is begun, 
and so on down to the gangway. Both sides of the pillar are 
worked similarly and at the same time toward the chute. 











232 


MINING 



Small cats, or buggies, are used to convey the coal from the 
working faces along the headings to the chute, where it is run 
down to the gangway below and loaded into the regular mine 
cars. This system affords a great degree of safety to the work¬ 
men, because whenever any signs of a fall of roof or coal occur, 
the men can reach the heading in a very few seconds and be 
perfectly safe. A great deal of narrow work must be done 
before any great quantity of coal can be produced. The breasts 
are driven in pairs and at intervals, to get a fair quantity of coal 
while the narrow work is being done, and they are not an essen¬ 


tial part of the system. It is claimed that the facility and 
cheapness with which the coal can be mined, handled, and 
cleaned in the mine more than counterbalance the extra expense 
for the narrow work. 

Battery Working.—Fig. 2 shows a method of opening a 
breast by two chutes c, c, when there is a great amount of refuse, 
or when a great amount of gas is given off. The chutes are 
extended up along the rib to within a few feet of the working 
face, either by planking carried on upright posts, or by building 
a jugular manway, as shown in Fig. 3, (o) and (6). These 
chutes, built of jugulars or inclined props and faced by 2-in. 


Fig. 2 






MINING 


233 


plank, are made as nearly air-tight as possible, to carry the air 
from the heading a to the working face. Fig. 2 shows a breast 
on a pitch too steep to enable the miner to keep up to the face. 
In seams of less than 35°, the platform / shown near the face of 
the breast is unneccessary, and in seams thicker than 12 ft. it 
cannot be built; hence, this method of working is applicable 
only to beds pitching more than 35°, and to thin seams. 

The coal is separated from the refuse on the platform /, and 
is run down the manway chutes and loaded into the cars from a 
platform projecting into the gangway g. The refuse is thrown 
in the middle of the breast behind the platform. A certain 
amount of coal is kept on the 
platform to deaden the blow 
from the falling coal. The 
chutes are timbered when the 
character of the coal, requires 
it. This plan can also be 
employed in thick seams 
having a heavy dip, if there is 
enough refuse to fill the cen¬ 
ter of the breast so that the 
miner can work without the 
platform. 

Fig. 3 (a) is a section 
through p, Fig. 2, when jugu¬ 
lars a, are used to form the 
manways ft, along the sides of 
the breast; and (ft) is a section 
through the same line when upright posts a are used to sup¬ 
port the plank in forming the manways ft. The refuse in these 
cases only partially fills the gob. 

In working very thick seams on heavy dips, where there is not 
enough refuse to fill the middle of the breast, the miner has 
nothing to stand on, the platform being impracticable; there¬ 
fore, it is necessary to leave the loose coal in the breast. Loose 
coal occupies from 50% to 90% more space than coal in the 
solid. This surplus is drawn out through a central chute. If 
the roof is poor, the movement of the coal will not in this way 
cause it to fall and mix with the coal; and if the floor is soft. 



(a) 



(b) 
Fig. 3 




















234 


MINING; 


the jugulars, which are stepped into the floor, are not so liable 
to be unseated, closing the manway and blocking the ventila¬ 
tion. The surplus is sometimes sent down the manways, 
leaving the loose coal in the center of the breast undisturbed, 
until the limit is reached. 

Single-Chute Battery.—To prevent the coal from running 
out through the chutes, the opening into the breast is closed by a 
battery constructed by laying heavy logs across the openings, 
as shown at b, Fig. 4, or else built on props, as shown at b, Fig. 



5; a hole is left in the center, or at one side of the battery, 
through which the coal may be drawn. The battery closes all 
the openings into the breast, except the space occupied by the 
jugular manways, and is made air-tight, or as nearly so as 
possible, by a covering of plank. 

Fig. 4 is a plan and section of a breast opened up by a single 
chute. The plan A is taken on the line m n shown on the 
section B, which section is taken on the line / l shown on the 
plan A. The pitch is great and the seam is so thick that the 
breast must be kept full of loose coal for the men to work upon, 








MINING 


235 


the surplus being drawn off at the battery b and run into the car 
standing on the gangway g through the chute c. A manway w 
is made along each side of the breast, for the purpose of ventila¬ 
tion and affording a passage for the men to reach the working 
face. The heading a is used for an air-course between breasts. 
The main airway h is driven over the gangway g, where it will be 
well protected. 

By drawing the surplus coal through a central chute, the 
manways are not injured so much as when it is drawn off 


p 



4 A B 

Fig. 5 


through side chutes, as the coal will move principally along the 
middle of the breast. When the breast is worked up to its 
limit, all the loose coal is run out of the breast and the drawing 
back of the pillars is commenced, unless for some purpose they 
are allowed to stand for a time. 

Double-Chute Battery.—Fig. 5 shows a plan and section of 
double-chute breasts used in very thick seams having a heavy 
dip. The breasts are entered by two main coal chutes c, each 
of which is provided with a battery b, through which the coal is 
drawn. A manway chute m is driven up through the middle of 











236 


MINING 


the pillar for a few yards and is then branched in both directions 
until each branch (slant chute) intersects the foot of a breast 
near the battery b, as shown. The jugular manways n, are 
started at this point and continued up each side of the breast. 
The main airway h is driven in the solid through the stump A 
above the gangway. By driving the main gangway g against 
the roof, as shown, the pitch of the chute is lessened, and the 
loading chute c is more readily controlled. 

When the main gangway is not driven against the roof, a gate 
is placed in the chute below the check-battery, which enables 

the loader to properly 
handle the coal. Coal 
in excess of the amount 
necessary to keep the 
miner up to the face 
may be drawn through 
the main battery, or 
sent down the manway 
chute, from which it is 
loaded through an air¬ 
tight check-battery. 

The main chutes are 
usually 8 or 9 ft. wide, 
but sometimes only for 
the first 6 or 8 ft.; above 
this they are driven 
about 6 ft. square. The 
manway and slant chutes 



Fig. 6 

are also about 6 ft. square, which makes an easy passage. 

When the seam is not thick enough to carry the return airway 
h, Fig. 5, over the gangway, the chutes are driven up in the 
manner there shown for a distance of about 30 ft., where they 
intersect the airway. The breast is opened out just above the 
airway, a battery being built in the airway immediately above 
each chute. A manway is driven from the gangway up through 
the middle of the stump until it intersects the airway, and a 
trap door is placed at this point to confine the air. This man¬ 
way is made about 4 ft.X6 ft., or smaller, according to the 
amount of air needed. 










MINING 


237 


Fig. 6 shows a less complicated plan than Fig. 5. The main 
chutes n, are driven up to the heading c, from which the breast 
is opened out; a log battery is built at the top of each chute at 
the points marked a. The chutes are used for drawing the 
battery coal, and for receiving the manway coal, and are also 
used for traveling ways. A check-battery b is placed in the 
chute to prevent the air-current from taking a short cut from 
the gangway through the chute to the breast airways. This 
check-battery is of great assistance to the loader when the chute 
has a very steep pitch, as he can readily control the flow of coal 
through the diaw-hole. 



Fig. 7 


All these methods are open to the objection that in case 
of any accident to the breast manway, by which the flow of 
air, shown by the arrows, is obstructed, there is no means of 
isolating the breast in which the accident occurs, and the 
ventilation of all the breasts beyond it is entirely stopped. 
To overcome this, sometimes the pillar A, shown in left- 
hand breast, in Fig. 6, is left in each breast to protect the 
airway. 

Rock-Chute Mining.—Fig. 7 shows a section of two seams, 
separated by a few yards of rock. Chutes from 4^ to 7 ft. 
high and 7 to 12 ft. wide are driven in the rock from the gang¬ 
way or level g to the level l in the seam above, at such an angle 


238 


MINING 


that the coal will gravitate from the upper seam into the gang¬ 
way g. The working, otherwise, is similar to that previously 
described. 

Fig. 8 shows how one or more seams are worked by connecting 
them by a stone drift, or tunnel, driven horizontally across the 
measures, through which the coal from the adjacent seams is 
taken to the haulage-way leading to the landing at the foot of 
the slope or shaft. Tunnels are sometimes driven horizontally 



Fig. 8 


through the measures from the surface, so as to cut one or 
more seams above water level. 

The lower seam of coal is worked from a gangway or 
level l, connected by a tunnel, or stone drift t, to the level or 
gangway g, in the thick seam. The stone drift may be 
extended right and left to open seams above and below the 
thick seam. This tunnel, or stone drift, is never driven under a 
breast in the upper seam, but directly under the middle of the 
pillar. 

In the upper and thicker seam, when the coal is very hard, 
a breast b is worked to the limit and the loose coal nearly all run 








MINING 


239 


out through the chute s into the gangway g. The monkey 
gangway m is driven near the top as a return airway, and is 
connected to the upper end of the chute 5 by a level heading n, 
and to the main gangway g by a heading v. These headings 
are driven for the purpose of ventilation and to provide access 
to the battery in case the chute s should be closed. In the 
lower seam, the breast is still being worked upwards in the 
ordinary way. 


FLUSHING OF CULM 

From 15% to 20% of the coal taken out of an anthracite 
mine, according to the methods used in the past, became so fine 
in the course of preparation through the breaker that it could 
not be used or sold, and had to be piled away as refuse. Recently 
the coarser portions of these culm piles have been screened 
out and sold for use as steam sizes, while the finer part, together 
with the fine material from the breaker, has been carried back 
into the mines with water to fill the abandoned portions of the 
underground workings. 

This culm is carried through a system of conveyors to the 
hopper, usually an old oil barrel, and the stream of water is 
conducted into the same hopper by a 3-in. pipe. The culm is 
then carried by the water through a pipe from 4 to 6 in. in 
diameter, which passes into the mine through the shaft, bore 
hole, or other opening, thence along the gangways to the cham¬ 
bers through the cross-cuts, and to the point where it is desired 
to deposit the culm. The bottoms or outlets of the chambers 
to be filled are closed by board partitions fitted closely, or by 
walls of slate or mine rubbish. The culm, as it issues from the 
end of the pipe, takes a very flat slope, and it is carried a long 
distance by the water, which ultimately filters through the 
deposited culm to the lower portion of the mine, to be pumped 
to the surface. When the chamber is filled to the roof, the pipe 
is withdrawn and extended to the next place to be filled, and 
so on. 

The amount of water used depends on the distance to which 
the culm is carried and the slope of the pipe. From 1$ to 1} 
lb. of water is required to flush 1 lb. of culm to level and down¬ 
hill places; 3 to 6 lb. of water to 1 lb. of culm to flush up-hill 


240 


MINING 


for heights varying from 10 to 100 ft. above the level of the 
shaft bottom. Any elevation of the pipe very materially 
increases the amount of water necessary. 

To remove the pillars after the intervening breasts have been 
filled with culm, the face of the pillar along the gangway is 
attacked, and a road driven up through the pillar, splitting it, 
as shown in the accompanying illustration. This road may be the 
full width of the pillar, but in general it is necessary to leave the 

narrow stump of coal on either 
side to keep up the fine flushed 
material in the adjoining 
breasts. The thickness of this 
supporting coal depends en¬ 
tirely on the condition of the 
flushed material behind it. 


EXPLOSIVES 

Explosives are divided into 
two general classes: Low 
explosives or direct-exploding 
materials, and high explosives 
or indirect - exploding mate¬ 
rials that require a detonator. 
The characteristics of a good 
blasting explosive are: suffi¬ 
cient stability and strength, difficulty of detonating by mechanical 
shock, handy form, absence of injurious effects on the user. 
Gunpowder or black powder is a low explosive; its composition 
depends on the purpose for which it is to be used, but the ingre¬ 
dients commonly used are saltpeter, sulphur, and charcoal. The 
high explosives are a mixture of nitroglycerine with an absorb¬ 
ing material, the composition of which is such that, in addition 
to thoroughly and permanently absorbing the nitroglycerine, 
it is itself a gas-producing compound. 

Safety explosives, or as they are called, permissible explosives, 
are compounds intended for use in gaseous mines, and they are 
so constituted that they will ignite without producing the 
extremely high temperature given by ordinary explosives. 






MINING 


241 


Permissible explosives may be arranged in four classes, the clas¬ 
sification being based on the nature and proportions of the 
substances used in the manufacture. These classes are hydrated, 
ammonium-nitrate, organic nitrate, and nitroglycerine explosives. 
Permissible explosives are made by nearly all the manufacturers 
of blasting powder and a list is published each year by the 
United States Bureau of Mines giving the powders that have 
passed the test for permissible explosives with full instructions 
how they should be used and where manufactured. 

Charging Explosives.—No invariable rule can be laid down 
as to the diameter and length of cartridges to be used under 
any and all circumstances, nor the amount or grade of powder 
required for all kinds of work. Much depends on the good 
sense and judgment of the persons using the explosives. In 
blasting coal, slate, marble, granite, freestone, or any other 
material that it is desirable to obtain in large blocks, cartridges 
of small diameter should be used in wide bore holes, the charge 
being rolled in several folds of paper, to prevent its touching 
the sides of the bore holes. The intensity of action and the 
crushing effect of the explosive are thus lessened. The charge 
must fit and fill the bottom of bore and be packed solid. If 
holes are comparatively dry, slit the paper of the cartridges 
lengthwise with a knife, and as each is dropped into the hole, 
strike a wooden rammer on it with sufficient force to make the 
powder completely fill the bottom and diameter of the bore. 
Where water is not present, a more perfect loading is made by 
taking powder out of cartridge and dropping it in loosely, ram 
each 6 or 8 in. of the charge, using the paper of each cartridge 
as a wad, to take down any powder that may have stuck to the 
sides of the hole. If water is standing in the hole, do not break 
the paper of the cartridges and avoid ramming more than 
enough to settle the charge on the bottom, using cartridges of 
as large diameter as will readily run into the bore. 

When cartridges are used, the last cartridge placed in the hole 
should contain an electric exploder, or cap with fuse attached. 
When loose powder is used, a piece of cartridge 2 or 3 in. in 
length, with exploder or cap attached, should be pressed firmly 
on top of charge. Some blasters put an exploder or cap in the 
first cartridge used, placing remainder of charge on top. 


242 


MINING 


If a seam is found in the rock, remove the powder from the 
cartridges and push it into the seam and fire a cap beside it. 

This will open the 
seam so that a larger 
quantity of explosive 
can be used, and the 
rock broken without 
drilling. 

Tamping. — In deep 
holes, water makes a 
good tamping, but fine 
sand, clay, etc. are gen¬ 
erally used. Fill in for 
the first 5 or 6 in. carefully, so as not to displace cap and 
primer; then with a hardwood rammer pack balance of 
.material as solid as possible, ramming with the hand alone. 






Fig. 2 


A 

li 


and not using any form of hammer. Never use a metal 
tamping rod. 

Firing.—If the work is wet, or the charge used under water, 
waterproof fuse must be used, and the end of the 
fuse protected by applying bar soap, pitch, or 
tallow around the edge of the cap. Water must 
not be allowed to reach the powder in the fuse 
or the fulminate in the cap. Exploding by elec¬ 
tricity is best under water at great depth, as the 
pressure of water is so great that the water is 
forced through the fuse and it so prevents firing. 

Nitroglycerine explosives always require deto¬ 
nation by a cap or exploder in order to develop 
their full force. Fig. 1 illustrates the method of 
attaching such an exploder to the end of a fuse 
and the placing of it in the cartridge. The explo¬ 



Fig. 3 


ders are loaded with fulminate of mercury and are slipped over 
the end of the fuse, after which the upper end is crimped tightly 



























MINING 


243 


against the end of the fuse, as shown. (Miners sometimes bite 
the caps on the fuse with their teeth; this should never be allowed, 
for should one explode in a man’s mouth it may prove fatal.) 
In placing the cap or exploder into the dynamite or giant- 
powder cartridge, care should betaken that only about two-thirds 
of the cap is embedded 
in the material of the 
cartridge, for if the fuse 
has to pass through a 
portion of the material 
before reaching the cap, 
it may ignite the mate¬ 
rial, thus causing defla¬ 
gration of the cartridge 
in place of detonation. The fumes given off by high explosives 
are much worse in the case of deflagrating a cartridge. 

The electric exploder. Fig. 2, consists of wires A and B that 
carry the current to the exploder; cement D (usually sulphur) 
that protects the explosive compound C (usually mercury ful¬ 
minate), all of which is contained in a copper shell, and a 
small platinum wire E is heated by the passage of a current and 
ignites the explosive. Fig. 3 shows the method of placing a cap 
or an electric exploder in a cartridge of powder. When a num¬ 
ber of holes are exploded at one time, the electric exploders a.e 




Fig. 5 


connected in series, as shown in Fig. 4, for a small number of 
holes, and as in Fig. 5 for a larger number. The battery for 
furnishing the current of electricity is a magneto machine that 
is worked by pulling up or by depressing a handle or rack bar, 
or else by turning a crank. 







244 


MINING 


Blasting by Electricity.—To blast by electricity, drill the 
number of holes desired to be fired at one time; the depth and 
spacing of the holes will depend on the character of rock, size 
of drill holes, etc., the blaster, of course, using his judgment in 
this matter.- Load the hole in the usual manner, fitting one 
cartridge with a fuse (electric exploder) instead of cap and fuse. 
The fuse head is fitted into the bottom end of the cartridge, or 
into the middle of one side of the cartridge, where a hole has been 
punched with a pencil or small sharp stick to receive it; push 
the powder close around the fuse head. The fuse can then 
be held in position by tying a string around the cartridge and 
the fuse wires, binding the wires to the cartridge, as shown in 
Fig. 3, where A shows head of fuse, B the two fuse wires, C 
string used to tie wires to cartridge. Hitches should never be 
made in fuse wires, as the insulation of the wires may be 
injured and the current of electricity pass from one wire to the 
other, without passing through the cap, hazarding a misfire. 

The cartridge containing the fuse is put in on top of the 
charge by some blasters; by others, at bottom of the charge. 
The best place for it is in the center of the charge. In tamping 
the hole, great care must be taken not to cut the wires, injure 
the cotton covering of fuse wires, or to pull the fuse out of the 
cartridge. At least 8 in. of the fuse wire should project above 
the hole, to make connections. 

When all the holes to be fired at one time are tamped, separate 
the ends of the two wires in each hole, and, by the use of con¬ 
necting wire, join one wire of the first hole with one of the 
second, the other or free wire of the second with one of the 
third, and so on to the last hole, leaving a free wire at each end 
hole. All connections of wires should be made by twisting 
together the bare and clean ends; it is best to have the joined 
parts bright. This may be done by scraping off the cotton 
covering at the ends of the wires to be connected, say for 2 in., 
then rubbing the wire with a small hard stone. All connections 
should be well twisted. Bare joints in wire should never be 
allowed to touch the ground, particularly if the ground is wet. 
This can be prevented by putting dry stones under the joints. 

The charges having all been connected, the free wire of the 
first hole should be joined to one of the leading wires, and the 





MINING 


245 


free wire of the last hole to the other of the two leading wires. 
The leading wires should be long enough to reach a point at 
a safe distance from the blast, say 250 ft. at least. All being 
ready, but not until the men are at a safe distance, connect the 
leading wires, one to each of the projecting screws on the front 
side or top of the battery,-through each of which a hole is bored 
for the purpose, and bring the nuts down firmly on the wires. 
Take hold of the handle for the purpose, lift the rack bar (or 
square rod, toothed on one side) to its full length, and press it 
down, for the first inch of its stroke with moderate speed, but 
finishing the stroke with all force, bringing rack bar to the 
bottom of the box with a solid thud, and the blast will be made. 
Do not chum rack bar up and down. It is unnecessary and is 
harmful to the machine. One quick stroke of the rack bar is 
sufficient to make the blast. Never use fuses (exploders) made 
by different manufacturers in the same blast. Connecting 
wire should be of same size as the fuse wire; leading wire 
should be at least twice as large. Covering on wire should not 
strip or come off easily. 

Arrangement of Drill Holes.—The arrangement of drill holes 
for driving and sinking should be such as to permit the easy 
handling of the drills and also to minimize the number of holes 
and the weight of the explosive. Two distinct systems are 
in use: the center cut, by which a center core or key is first 
removed, and after that concentric layers about this core; the 
square cut, in which the lines of holes are parallel to the sides of 
the excavation, the rock being removed in wedges instead of in 
concentric circles. 

Thawing Dynamite.—All frozen cartridges should be thawed, 
as, when frozen, cartridges are very hard to explode, and even 
if they do explode, the results are not nearly as satisfactory as 
when properly thawed. When cartridges are frozen, they 
should not be exposed to a direct heat, but should be thawed 
by one of the following methods: (1) Place the number of 
cartridges needed for a day’s work on shelves in a room heated 
by steam pipes (not live steam) or a stove; where regular blast¬ 
ing is done, a small house can be built for this purpose, fitted 
with a small steam radiator. Exhaust steam through these 
pipes gives all heat necessary. The house should be banked 


246 


MINING 


around with earth, or, preferably, with fresh manure. (2) Use 
two water-tight kettles, one smaller than the other, put cart¬ 
ridges to be thawed in smaller kettle, and place it in larger 
kettle, filling space between the kettles with hot water at, say, 
130° to 140° F., or so that it can be borne by the hand. To 
keep water warm, do not try to heat it in the kettle, but add 
fresh warm water. Cover kettles to retain heat. In thawing 
do not allow the temperature to get above 212° F. (3) Where 
the number of cartridges to be thawed is small, they may be 
placed about the person of the blaster until ready for use, the 
heat of the body thawing the cartridges. 


MACHINE MINING 

There are four general types of mining machines in use; pick 
machines, chain-cutter machines, cutter-bar machines, and 
longwall machines. The first two are the types almost uni¬ 
versally used in America. Cutter-bar machines have almost 
entirely disappeared from use. Longwall mining machines have 
not been very generally adopted in America, as the longwall 
method of mining is not extensively used. Both compressed 
air and electricity are used for operating mining machines. 

Pick machines work very similarly to a rock drill. They can 
be used wherever mining machines are applicable, and their 
particular advantage is that they are more perfectly under the 
control of the operator, who can cut around pyrites and similar 
obstructions without cutting them with the machine. This 
renders such a machine particularly applicable for seams of 
coal having rolls in the bottom and containing pyrites or other 
hard impurities. They are also applicable for working pillars 
on which there is a squeeze, as they are light and can be easily 
handled and readily removed. 

A good pick machine will undercut 450 sq. ft. in 10 hr., while 
an ordinary miner will undercut 120 sq. ft. in the same time. 
In a seam varying from 4§ to 6 ft. in thickness, the machine 
will undercut from 50 to 100 T. of coal in 10 hr. The cost of 
undercutting under these conditions has been given as approxi¬ 
mately 10c. per T. Extraordinary records show 1,400 sq. ft. 



MINING 


247 


to have been cut in 9 hr. in Western Pennsylvania, ana in an 
8 ft. seam, 240 T. have been undercut in a shift of 10 hr. 

Chain-cutter machines consist of a low metal bed frame upon 
which is mounted a motor that rotates a chain to which suitable 
cutting teeth are attached. To operate chain machines to the 
best advantage, the coal should be comparatively free from 
pyrites. They also require more room than pick machines, 
and a space from 12 to 15 ft. in width is necessary along the 
face to work them to advantage. A good chain cutter will make 
from 30 to 45 cuts, 44 in. wide and 6 ft. deep, in 10 hr. under 
moderately fair conditions, while in high seams two men hand¬ 
ling the same machine under ordinary conditions can make 60 
cuts per shifty and under particularly favorable conditions, 80 
to 120 cuts per shift. 

Shearing.—All the pick machines can be converted into 
shearing machines and can be used for longwall work by using a 
longer striking arm and a longer supply hose. The chain 
machines are used to do shearing work by having the cutting 
parts turned vertically. 


VENTILATION OF MINES 

COMPOSITION AND MEASUREMENT OF AIR 

Air consists chiefly of oxygen and nitrogen, with small and 
varying amounts of carbonic-acid gas, ammonia gas, and 
aqueous vapor. These gases are not chemically combined, 
but exist in a free state in uniform proportion of 79.3% nitrogen 
and 20.7% oxygen by volume and 77% nitrogen and 23% oxy¬ 
gen by weight. Wherever air is found, its composition is 
practically the same. The weight of 1 cu. ft. of air at 32° F. 
and under a barometric pressure of 30 in. is .080975 lb. Air 
decreases in weight per cubic foot as its height above the sea 
level increases and it increases in weight below that level. 

The weight of 1 cu. ft. of dry air at any temperature and 
barometric pressure is found by the formula 

1.3253 X B 

w = -, 

459 -M 



248 


MINING 


in which w — weight of 1 cu. ft. of dry air; 

B = barometric pressure, in inches of mercury; 
t = temperature, in degrees F. 

The constant 1.3253 is the weight in pounds avoirdupois of 
459 cu. ft. of dry air at a temperature of 1° F. and 1 in. baro¬ 
metric pressure. 

Example. —Find the weight of 1 cu. ft. of dry air at a tem¬ 
perature of 60° F. and a barometric pressure of 30 in. 

Solution. —Applying the formula 


1.3253X30 

459+60 


= .0766 lb. 


The atmospheric pressure is the pressure caused jpy the weight 
of the atmosphere above a given point. It is measured by the 
barometer, and this gives rise to the synonymous term baro¬ 
metric pressure. Atmospheric pressure is usually stated in 
pounds per square inch, while barometric pressure is stated 
in inches of mercury. Thus, at sea level, the atmospheric 
pressure under normal conditions of the atmosphere is 14.7 lb. 
per sq. in., while the barometric pressure at the same level is 
30 in. of mercury column, or simply 30 in. 

Mercurial Barometer.—The mercurial barometer is often 
called the cistern barometer; or, when the lower end of the tube 
is bent upwards instead of the mouth of the tube being sub¬ 
merged in a basin, it is known as the siphon barometer. The 
instrument is constructed by filling a glass tube 3 ft. long, and 
having a bore of j in. diameter, with mercury, which is boiled 
to drive off the air. The thumb is then placed tightly over 
the open end, the tube inverted, and its mouth submerged in 
a basin of mercury. When the thumb is withdrawn, the mer¬ 
cury sinks in the tube, flowing out into the basin, until the 
top of the mercury column is about 30 in. above the surface 
of the mercury in the basin, and after a few oscillations above 
and below this point, comes to rest. The vacuum thus left 
in the tube above the mercury column is as perfect a vacuum 
as it is possible to form, and is called a Torricelli vacuum, after 
its discoverer. 

Aneroid Barometer.— The aneroid barometer is a more port¬ 
able form than the mercurial barometer. It consists of a 



MINING 


249 


brass box resembling a steam-pressure gauge, having a similar 
dial and pointer, the dial, however, being graduated to read 
inches, corresponding to inches of mercury column, instead of 
reading pounds, as in a pressure gauge. Within the outer case 
is a delicate brass box having its upper and lower sides corru¬ 
gated, which causes it to act as a bellows, moving in and out 
as the atmospheric pressure on it changes. The air within 
the box is partially exhausted, to render it sensitive to atmos¬ 
pheric changes. The movement of the upper surface of the 
box is communicated to the pointer by a series of levers, 
and the dial is graduated to correspond with the mercurial 
barometer. 

Calculation of Atmospheric Pressure.—The weight of the 
mercury column of the barometer is the exact measure of the 
pressure of the atmosphere, since it is the downward pressure 
of the atmosphere that supports the mercury column, area 
for area; that is, the pressure of the atmosphere on 1 sq. in. 
supports a column of mercury having an area of 1 sq. in., 
and whose height is such that the weight of the mercury col¬ 
umn is equal to the weight of the atmospheric column. Hence, 
as 1 cu. in. of mercury weighs .49 lb., the atmospheric pressure 
that supports 30 in. of mercury column is .49X30=14.7 lb. 
per sq. in. In like manner, the atmospheric pressure corre¬ 
sponding to any height of mercury column may be calculated. 
The sectional size of the mercury column is not important 
because it is supported by the atmospheric pressure on an 
equal area, but the calculation of pressure is based on 1 sq. in. 

Water Column Corresponding to Any Mercury Column.—The 
density of mercury referred to water is practically 13.6; hence, 
the height of a water column corresponding to a given mercury 
column is 13.6 times the height of the mercury column. For 
example, at sea level, where the average barometric pressure 
is 30 in. of mercury, the height of water column that the 
atmospheric pressure will support is 13.6X?? = 34 ft. This is 
the theoretical height to which it is possible to raise water by 
means of a suction pump, but the length of the suction pipe 
should not exceed 75% or 80% of the theoretical water column. 

Finding Depth of Shafts.—The barometer is often used to 
determine the depth of a shaft or the depth of any point in a 


250 


MINING 


mine below a corresponding point on the surface. The aneroid 
is used for this work, being more portable. Allowance must 
always be made in such cases for the ventilating pressure of 
the mine. A simple formula often used for such calculations 
is the following: 

// = 55,000 

in which H = difference of level between two stations, in feet; 

r — reading of barometer at higher station, in inches; 

R = reading of barometer at lower station, in inches. 

The most important use of the barometer in mining practice, 
however, is found in the warning that it gives of the decrease 
of atmospheric pressure, and the expansion of mine gases that 
always follows. 

GASES FOUND IN MINES 

Oxygen, O, is a colorless, odorless, tasteless, non-poisonous 
gas. It is heavier than air, having a specific gravity of 1.1056. 
It is the great supporter of life and combustion. 

Nitrogen, N, is a colorless, odorless, and tasteless gas; it is 
neither combustible nor a supporter of combustion. It is 
not poisonous, and is lighter than air, having a specific gravity 
of .9713. Nitrogen is a particularly inert gas; it takes no active 
part in any combustion, in the sense of causing such combustion. 
Its province is to dilute oxygen of the atmosphere, on which 
life depends. 

Methane, CHa, often called light carbureted hydrogen, or 
marsh gas, is a chemical compound, consisting of 4 atoms of 
hydrogen to 1 atom of carbon. It is one of the chief gases 
occluded in coal seams, and results from the metamorphism 
of the carbonaceous matter from which coal is formed, when 
such metamorphism has taken place with the exclusion of 
air, and in presence of water. Pure methane is colorless, 
odorless, and tasteless, and is lighter than air. Its specific 
gravity is .559, and it diffuses rapidly in the air, forming a 
firedamp mixture. Marsh gas burns with a blue flame, but 
it will not support combustion, and a lamp placed in it is 
immediately extinguished 

Carbon monoxide, CO, often called carbonic oxide gas, or while- 
damp, is a chemical compound consisting of 1 atom of carbon 



MINING 


251 


united to 1 atom of oxygen. To a certain extent it occurs as 
an occluded gas in coal. It is chiefly formed, however, in coal 
mines, by the slow combustion of carbonaceous matter in the 
gobs or waste places of the mine; where the supply of air is 
limited. It is always the product of the slow combustion of 
carbon in a limited supply of air, and is therefore one of the 
chief products of gob fires; it is also a product of the explosion 
of powder in blasting. This gas often fills the crevice made 
behind a standing shot, and causes the flash that takes place 
when the miner puts his lamp behind such shot to examine the 
same. This gas is formed in large quantities whenever the 
flame of a blast or explosion is projected into an atmosphere 
in which coal dust is suspended. The force of a blast often 
blows the dust into the air, and the flame acting on it distils 
carbon monoxide. 

Carbon monoxide is lighter than air, having a specific gravity 
of .967, and it therefore accumulates near the roof and in the 
higher working places. It is colorless, odorless, and tasteless, 
but is combustible, burning with a light-blue flame. It is the 
flame often seen over a freshly fed anthracite fire. Carbon 
monoxide is a very poisonous gas, and acts on the human 
system as a narcotic, producing drowsiness or stupor, followed 
by acute pains in the head, back, and limbs, and afterward 
by delirium. When breathed into the lungs, it absorbs the 
oxygen from the blood, or, in other words, poisons the blood. 

Carbon monoxide is detected in mine workings by its effect 
on the flame of a lamp, which burns more brightly in the pres¬ 
ence of the gas, and reaches upwards as a slim, quivering taper, 
having often a pale-blue tip that, however, is not readily 
observed. 

Carbon dioxide, CO 2 , often called carbonic-acid gas, black- 
damp, or choke damp, is a chemical compound consisting of 
1 atom of carbon united to 2 atoms of oxygen. It is heavier 
than air, having a specific gravity of 1.529. It therefore 
accumulates near the floor or in the low places of the mine 
workings. It is always the result of the complete combustion 
of carbon, in a plentiful supply of air, and is a product of the 
breathing of men and animals, burning of lamps, or any other 
complete combustion. It is always present in occluded gases. 



252 


MINING 


Carbon dioxide is a colorless, odorless gas, but possesses a 
peculiarly sweet taste, which may be detected in the mouth 
when it is inhaled in large quantities. It is not combustible, 
nor is it a supporter of combustion. Lamps are at once extin¬ 
guished by it. It diffuses slowly into the atmosphere, and is a 
difficult gas to remove in ventilating. It is not poisonous, but 
suffocates by excluding oxygen from the lungs. Its effect, 
when breathed for any length of time, is to cause headache and 
nausea, followed by weakness and pains in the back and limbs; 
when present in larger quantities, it causes death by suffoca¬ 
tion. This gas, when present in the firedamp mixtures, has 
the opposite effect from that of carbon monoxide, inasmuch 
as it narrows the explosive range of the firedamp, and ren¬ 
ders such mixtures inexplosive, which would otherwise be 
explosive. 

Carbon dioxide is detected in the mine air by the dimness of 
the lamps and by their extinguishment when the gas is present 
in larger quantities. It should always be looked for at the 
floor, and in low places of the mine workings. 

Hydrogen sulphide , HiS, occurs at times as an occluded gas in 
coal seams, but more often exudes from the strata immediately 
underlying or overlying those seams. It is generally supposed 
to be formed by the disintegration of pyrites in the presence 
of moisture. It is heavier than air, having a specific gravity 
of 1.1912. It is a colorless gas, having a very disagreeable 
odor resembling that of rotten eggs, and is known to the miners 
as slinkdamp. It is an exceedingly dangerous gas when 
occurring in considerable quantities. When mixed with 7 times 
its volume of air, it is violently explosive. It is extremely 
poisonous, acting to derange the system, when breathed in 
small quantities, and, when inhaled in larger quantities, it pro¬ 
duces unconsciousness and prostration. Its smell serves as 
the best means for its detection. 

The general term firedamp relates to any explosive mixture 
of marsh eras and air, although in some localities this term is 
understood as referring to any mixture of methane and air 
whatever, whether explosive or otherwise. Many persons 
speak of pure methane as firedamp. The first meaning given, 
however, is the general acceptance of the term. 


MINING 


253 


The term afterdamp relates to the gaseous mixture that 
exists in mine workings after an explosion of gas or coal dust. 
The composition of af¬ 
terdamp is exceedingly 
variable, and admits of 
no general analysis that 
can be applied with cer¬ 
tainty to any one explo¬ 
sion. The conditions 
that obtain in an explo¬ 
sion are so manifold, 
and control so complete¬ 
ly the character of the 
gases formed, that it is 
impossible to give more than a general analysis of afterdamp. 

Outbursts of gas are frequent occurrences in some coal seams. 
They are caused by the occluded gas finding its way to a vertical 
crevice or cleat in the coal seam, as illustrated in Fig. 1, and the 
pressure of the gas thus becomes distributed over a large area. 

Testing for Gas by Lamp Flame.—Methane and firedamp 
are detected in mine workings by the small flame cap that envel¬ 
opes and surmounts the flame of the lamp in a firedamp mixture. 

o & 



Fig. 2 

This flame cap is caused by the gaseous mixture, which bums 
as it comes in contact with the flame. The proportion of gas 
in the mixture determines the height of the flame cap. 

























254 


MINING 


In Fig. 2, the heights of flame cap due to the presence of 
different proportions of methane are shown. These heights, 
as given, refer to the experimental heights of flame cap obtained 
with pure methane. It should be observed, however, that the 
presence of other gases in the firedamp will vary its explosive 
character, and this fact very materially modifies the explosive¬ 
ness of certain caps. 

Constants for Mine Gases.—The following table shows the 
symbols, specific gravities, and relative velocities of diffusion 
and transpiration of the principal mine gases, arranged in the 
order of their specific gravities, air being taken as 1. The 
values given in the next to the last column were obtained by 
experimenting with the gases, and agree quite closely with 
the calculated values given in the preceding column. This 
column shows that 1,344 volumes of methane will diffuse in 
the same time as 1,000 volumes of air, or 812 volumes of car¬ 
bon dioxide. 

TABLE OF MINE GASES 


Name of Gas 

Sym¬ 

bol 

Specific 

Gravity 

1 

Vsp. gr. 

Relative 
Velocity 
of Dif¬ 
fusion 

(Air = 1) 

Relative 
Velocity 
of Trans¬ 
piration 

(Air = 1) 

Air. 


1.00000 

1.0000 

1.000 

1.0000 

Carbon dioxide.... 

CO 2 

1.529 

.8087 

.812 

1.2371 

Hydrogen sulphide 

H 2 S 

1.1912 

.9162 

.95 


Oxygen. 

0 

1.1056 

.9510 

.9487 

.903 

Olefiant. 

C 2 H 4 

.978 

1.0112 

1.0191 

1.788 

Nitrogen. 

N 

.9713 

1.0147 

1.0143 

1.0303 

Carbon monoxide.. 

CO 

.967 

1.0169 

1.0149 

1.034 

Steam. 

II 2 O 

.6235 

1.2664 



Methane. 

CH\ 

.559 

1.3375 

1.344 

1.639 

Hydrogen. 

H 

.06926 

3.7794 

3.83 

2.066 


SAFETY LAMPS 

The safety lamp is designed to give light in gaseous workings 
without the danger of igniting the gases present in the atmos¬ 
phere. Its principle depends on the cooling effect that an 
iron-wire gauze exerts on flame. It is well known that all 





























MINING 


255 


gases ignite at certain fixed temperatures, and if this tempera¬ 
ture is decreased from any cause, the flame is extinguished. 
Safety lamps are also used for testing for gas. 

The essential features of a lamp designed for general mine 
work are: safety in strong currents, good illuminating power, 
security for lock fastening, freedom from flaming, security 
against accident, simplicity of construction. The essential 
features of a lamp for testing purposes are: free admission of 
air below the flame, no reflecting surface behind the flame, 
ability to test for a thin layer of gas at the roof. The Davy 
lamp in the hands of a careful person may be made to detect 
the presence of gas in quantities as low as 3%. It is claimed 
by some fire-bosses that 2% of gas may be detected with a 
good Davy. For the detection of small quantities of gas, 
especially constructed lamps have been used. 

Types of Safety Lamps.—In the year 1815, Sir Humphrey 
Davy and George Stevenson, the latter a poor miner, discov¬ 
ered simultaneously, that flame would not pass through small 
openings in a perforated iron plate. This led to the construc¬ 
tion of what are known as the Davy and the Stevenson or 
“Geordy,” lamps. The Davy lamp is still a great favorite 
among fire-bosses for the detection of gas in mine air. Inas¬ 
much as all safety lamps, of which there are a large number, 
depend on the same principle, only such lamps as possess 
essential features, and show important improvements and the 
gradual developments in safety-lamp construction are here 
mentioned. They are the Davy, Clanny, Mueseler, Marsant 
Ashworth-Hepplewhite-Gray, and Wolf. 

Oils for Safety Lamps.—Most safety lamps burn vegetable 
oils, which are considered the safest for mining use; such oils 
are rape-seed oil and colza oil, made from cabbage seed. Seal 
oil is also largely used. Seal oil affords a better light than 
vegetable oils, and in its use there is less charring of the wick. 
A mixture of 1 part of coal oil to 2 parts of rape or seal oil is 
often used and improves the light, but the smoke from the 
flame is increased. The Ashworth-Hepplewhite-Gray lamp 
is constructed to burn coal oil, or a mixture of coal and lard 
oil. The Wolf lamp is especially designed for burning naphtha 
or benzine. Special tests have been made to prove the safety 


256 


MINING 


of using such a fluid in this lamp, and resulted in demonstrat¬ 
ing the fact that the lamp was safe under any conditions that 
might arise. A thorough test was made, the oil vessel of the 
burning lamp being heated to 180° F., at which point the lamp 
was extinguished without manifesting any dangerous results. 

Locking Safety Lamps.—The ordinary lock consists of a 
lead plug, which, when inserted in the lamp, will show the 
least tampering on the part of the miner. Other locks consist 
of an ordinary turnbolt operated by a peculiar key. Magnetic 
locks allow of the opening of the lamp only by means of a 
strong magnet kept in the lamp room. 

Cleaning Safety Lamps.—Safety lamps should be thoroughly 
and regularly cleaned and filled between each shift. Each 
lamp should then be lighted and inspected by a competent 
person before being given to the miner. A careful inspection 
of the gauze of the lamp is necessary, as well as of all the joints 
by which air may enter the lamp. It should be known to a 
certainty that each lamp is securely locked before it leaves the 
lamp room. 

Relighting Stations.—The relighting stations are located at 
certain places in gaseous mines where they can be supplied 
with a current of fresh air, and where there is no danger from 
the gases of the mine. The lamp is apt to be overturned, or 
to fall, and is often extinguished thereby; and if these stations 
were not provided, the man would have to return with his 
lamp to the surface in order to have it relighted. Such a 
station is always located at the entrance of the gaseous portion 
of a mine, in cases where the entire mine does not liberate gas. 
A number of self-igniters have been invented and some are 
used. If the lamp goes out all that is needed is to turn a screw 
in the bottom of the lamp and a spark is made which relights 
the lamp. 

Illuminating Power of Safety Lamps.—The accompanying 

table gives the illuminating power or candlepower of some of 
the principal lamps. The light of a sperm candle is taken 
as 1, or unity. 

Acetylene Mine Lamps.—Acetylene gas is generated by 
dropping water on to calcium carbide; as in the automobile 
and bicycle acetylene lamps. The miner’s lamps using 


MINING 


257 


acetylene are small brass lamps consisting of two main parts, 
the carbide container and the water tank; a regulator limits 
the amount of water dropping on to the carbide. The gas 
given off burns with a bright white flame. Besides giving 
greater illumination, acetylene lamps bum practically without 
generating soot, and are much less harmful to the miner’s 
respiratory organs than the constantly smoking oil lamps. 
Ventilation is also facilitated, owing to the acetylene lamp 
consuming less oxygen than any other. Acetylene illumination 
is also cheaper than oil lighting. 

Electric Mine Lamps.—Several forms of electric mine lamps 
have been invented. Some of them are in use by miners, but 
others are not extensively used, on account of their weight. 
These heavier types may be used by mine rescue parties. 
Electric mine lamps should have a battery that is not heavy, 
a good tungsten lamp, and a good strong reflector. In the 
Hirsch is an example which is carried by the miner, the battery 
is fastened on the miner’s back to a belt passing around his 
waist. An insulated eopper wire transmits the current from 

LIGHT GIVEN BY SAFETY LAMPS 


Name of Lamp 

Illuminating Power 
of Lamp 

Candlepower 

Davy. 

.16 

Geordy. 

.10 

Clanny. 

.20 

Mueseler. 

.35 

Evan Thomas. 

.45 

Marsaut, 3 gauzes. 

.45 

Marsaut, 2 gauzes. 

.55 

Marsaut, with Howat’s deflector. 

.65 

Ashworth-Hepplewhite-Gray. 

.65 

Wolf....'. 

.90 



the battery to the lamp on the miner’s cap. The battery is 
charged by connecting it with the current in the lamp house 
at night and is ready for use the next morning. A type of 
electric amps to carry in the hand and for use on locomotives, 
is the Hubbell electric lantern 

















258 


MINING 


EXPLOSIVE CONDITIONS IN MINES 

In the ventilation of gaseous seams, the air-current may be 
rendered explosive by the sudden occurrence of any one of a 
number of circumstances that cannot be anticipated. Hence, 
the condition of the air-current should be maintained far 
within the explosive limit. Among these are the following: 
(1) Derangement of the ventilating current; (2) sudden 
increase of gas due to outbursts, falls of roof, feeders, fall of 
barometric pressure, etc.; (3) Presence of coal dust thrown into 
suspension in the air, in the ordinary working of the mine, or 
by the force of blasting at the working face, or by a blown-out, 
or windy, shot; (4) Pressure due to a heavy blast, or any con¬ 
cussion of the air caused by closing of doors, etc.; (5) rapid 
succession of shots in close workings; (6) accidental discharge 
of an explosive in a dirty atmosphere. The explosive condi¬ 
tions vary considerably in different coal seams, as the nature 
of the coal and its enclosing strata, its friability and inflam¬ 
mability, together with the character of its occluded gases, 
determine, to a large extent, the explosive conditions. A 
great many of these conditions have been investigated by the 
U. S. Bureau of Mines and the results published in pamphlet 
form. 

Mine Explosions.—The explosion of gas in a mine usually 
arises from the ignition of an explosive mixture of gas and air 
called firedamp, which has accumulated in some unused 
chamber or cavity of the roof, or in the waste places of the 
mine, and has been ignited by a naked light, by the flame of a 
shot, or by a mine fire. The initial force of an explosion is 
generally expended locally, but the flame continues to feed 
upon the carbon monoxide generated by the incomplete com¬ 
bustion of the firedamp mixture, and distilled also from the 
coal dust thrown into the air by the agitation. Air is required 
to burn this carbon monoxide; this causes the flame to travel 
against the air-current, or in the direction in which fresh air is 
found. In the other direction, or behind the explosion, the 
flame is soon extinguished in its own trail when the initial 
force of the explosion is expended. The explosion continues 
to travel along the airways against the current as long as there 
is sufficient gas or coal dust for it to feed upon, or until its 


MINING 


259 


temperature is cooled below the point of ignition, by some 
cause such as, for example, the rapid expansion of the area of 
the workings. The chief factor in transmitting an explosion 
is the presence of carbon monoxide, which lengthens the flame 
and extends the effect. 

The recoil of an explosion is the return of the flame along the 
path that it has just traversed. In the recoil, the flame burns 
more quietly, advances more slowly and travels close to the roof. 

QUANTITY OF AIR REQUIRED FOR VENTILATION 

The quantity of air required for the adequate ventilation 
of a mine cannot be stated as a rule applicable in every case. 
Regulations that would supply a proper amount of air for the 
ventilation of a thick seam would cause great inconvenience 
if applied without modification to the workings in a thin seam. 

The quantity of air required by the laws of the several States 
is generally specified as 100 cu. ft. per min. per man and in many 
cases an additional amount of 500 cu. ft. per min. per animal 
is stated. This quantity is in no case stated as the actual 
amount of air required for the use of each man or animal, 
but is only the result of experience, as showing the quantity 
of air required for the proper ventilation of the average mine, 
based on the number of men and animals employed. The 
number of men employed in a mine is an indication of the 
extent of the working face, while the number of animals 
employed is an indication likewise of the extent of the haulage 
roads, or the development of the mine. These amounts refer 
particularly to non-gaseous seams. 

The Bituminous Mine Law of Pennsylvania specifies that 
there shall be not less than 100 cu. ft. per min. per person in 
any mine, while 150 cu. ft. is required in a mine where fire¬ 
damp has been detected. 

The Anthracite Mine Law of Pennsylvania specifies a mini¬ 
mum quantity of 200 cu. ft. per min. per person. Each of 
these laws contains modifying clauses, which specify that the 
amount of air in circulation shall be sufficient to “dilute, 
render harmless, and sweep away” smoke and noxious or 
dangerous gases. 


260 


MINING 


ELEMENTS OF VENTILATION 

The elements in any circulation of air are (1) horsepower, or 
power applied; (2) resistance of airways, or mine resistance, 
which gives rise to the total pressure in the airway; (3) velocity 
generated by the power applied against the mine resistance. 

Horsepower or Power of Current.—The power applied is often 
spoken of as the power upon the air. It is the effective power 
of the ventilating motor, whatever this may be, including all 
the ventilating agencies, whether natural or otherwise. The 
power upon the air may be the power exerted by a motive 
column due to natural causes, or to a furnace, or may be the 
power of a mechanical motor. The power upon the air is 
always measured in foot-pounds per minute, which expresses 
the units of work accomplished in the circulation. 

Mine Resistance.—The resistance offered by a mine to the 
passage of an air-current, or the mine resistance, is due to the 
friction of the air rubbing along the sides, top, and bottom of 
the air passages. This friction causes the total ventilating 
pressure in the airway, and is equal to it, or, the total pressure 
is equal to the mine resistance or 

R — pa, 

in which R = resistance; 

p — unit of ventilating pressure; 
a = sectional area of airway. 

Velocity of Air-Current.—Whenever a given power is applied 
against a given resistance, a certain velocity results. For 
example, if the power u, in foot-pounds per minute, is applied 
against the resistance pa, a velocity of v, in feet per minute, 
is the result; and as the total pressure p a moves at the velocity 
of v, the work performed each minute by the power applied is 
the product of the total pressure by the space through which 
it moves per minute, or the velocity. Thus, u—(p a)v. 

Relation of Power, Pressure, and Velocity.—The relation 
of power, pressure, and velocity is not a simple one. For 
example, a given power applied to move air through an airway 
establishes a certain resistance and velocity in the airway. 
The resistance of the airway is not an independent factor; that 
is, it does not exist as a factor of the airway independent of the 
velocity, but bears a certain relation to the velocity. Power 


MINING 


261 


always produces resistance and velocity, and these two factors 
always sustain a fixed relation. 

This relation is expressed as follows: The total pressure or 
resistance varies as the square of the velocity; i. e., if the power 
is sufficient to double the velocity, the pressure will be increased 
4 times; if the power is sufficient to multiply the velocity 
3 times, the pressure will be increased 9 times. Thus, a change 
of power applied to any airway means both a change of 
pressure and a change of velocity. 

Again, as the power is expressed by the equation u=(p a)v, 
and as p a, or the total pressure, varies as v 2 , the work varies 
as v 3 . Therefore, if the velocity is multiplied by 2, and, conse¬ 
quently, the total pressure by 4, the work performed ( p a)v will 
be multiplied by 2 3 = 8, or the power applied varies as the 
cube of the velocity. 

MEASUREMENT OF VENTILATING CURRENTS 

The measurement and calculation of any circulation in a 
mine airway includes the measurement of: (1) the velocity 
of the air-current, (2) of pressure, (3) of temperature, (4) cal¬ 
culation of pressure, quantity, and horsepower of the circula¬ 
tion. These measurements should be made at a point on the 
airway where the airway has a uniform section for some dis¬ 
tance, and not far from the foot of the downcast shaft or the 
fan drift. 

Measurement of Velocity.—For the purpose of mine inspec¬ 
tion, the velocity of the air-current should be measured at the 
foot of the downcast, at the mouth of each split of the air-cur¬ 
rent, and at each inside breakthrough, in each split. These 
measurements are necessary in order to show that all the air 
designed for each split passes around the face of the workings. 

The measurement of the velocity of a current is best made by 
means of the anemometer. 

Rule. —To obtain the quantity of air passing in cubic feet per 
minute, multiply the area of the airway, at the point where the 
velocity is measured, by the velocity. 

Measurement of Pressure.—The measurement of the ven¬ 
tilating pressure is made by means of a water column in the 
form of a water gauge, which is simply a glass U tube open at 


262 


MINING 


both ends, as shown in Fig. 3. Water is placed in the bent 
portion of the tube, and stands at the same height in both 
arms of the tube when each end of the tube is subjected to the 
same pressure. If, however, one end of the tube is subjected 
to a greater pressure than the other end, the water will be 
forced down in that arm of the tube, and will rise a correspond¬ 
ing height in the other arm, the difference of level in the two 
arms of the tube representing the water column balanced by 
the excess of pressure to which the water in the first arm is 
subjected. An adjustable scale, graduated in inches, measures 




Fig. 4 


Fig. 3 

the height of the water column. The zero is adjusted to the 
lower water level and the upper water level will then give the 
reading of the water gauge. One end of the glass tube is 
drawn to a narrow opening to exclude dust, while the other 
end is bent to a right angle, and passing back through the 
standard to which the tube is attached, is cemented into the 
brass tube that passes through a hole in the partition or brat¬ 
tice, when the water gauge is in use. The bend of the tube 
is contracted to reduce the tendency to oscillation in the height 
of water column. 











































MINING 


263 


When in use, the water gauge must be in a perpendicular 
position. It is placed upon a brattice occupying a position 
between two airways, as shown at A, Fig 4. The brass tube 
forming one end of the water gauge is inserted in a cork, and 
passes through a hole bored in the brattice. The water gauge 
must not be subjected to the direct force of the air-current, as 
in this case the true pressure will not be given. Fig. 4 shows 
the instrument occupying a position in the breakthrough, 
between two entries. It will be observed that the water 
gauge records a difference of pressure, each end of the water 
gauge being subject to atmospheric pressure, but one end in 
addition being subject to the ventilating pressure, which is the 
difference of pressure between the two entries. The water 
gauge thus permits the measurement of the resistance of the 
mine inbye from its position between two airways. If placed 
in the first breakthrough, at the foot of the shaft, it measures 
the entire resistance of the mine, but if placed at the mouth 
of a split, it measures only the resistance of that split. It 
never measures the resistance outbye from its position in the 
mine, but always inbye. 

Measurement of Temperature.—It is important to measure 
the temperature of the air-current at the point where the veloc¬ 
ity is measured, as the temperature is an important factor of 
the volume of air passing. 

Calculation of Mine Resistance.—The mine resistance is 
equal to the total pressure pa that it causes. This mine resist¬ 
ance is dependent on three factors: (1) The resistance k 
offered by 1 sq. ft. of rubbing surface to a current having a 
velocity of 1 ft. per min. The coefficient of friction k, or the 
unit of resistance, is the resistance offered by the unit of rubbing 
surface to a current of a unit velocity. This unit resistance 
has been variously estimated by different authorities. The 
value most universally accepted, however, is that known as the 
Atkinson coefficient .0000000217. (2) The mine resistance, 

which varies as the square of the velocity. (3) The rubbing 
surface. Hence, if the unit resistance is multiplied by the 
square of the velocity, and by the rubbing surface, the total 
mine resistance as expressed by the formula pa = ksv 2 , will be 
obtained. 


264 


MINING 


Calculation of Power, or Units of Work per Minute.—If the 

total pressure is multiplied by the velocity in feet per minute, 
with which it moves, the units of work per minute, or the power 
upon the air, will be obtained. Hence, u = pav = ksv 3 , which is 
the fundamental expression for work per minute, or power. 

The Equivalent Orifice. —The term equivalent orifice, often 
used in regard to ventilation, evaluates the mine resistance, or, 
as will be seen from the equation given for its value, it expresses 
the ratio that exists between the quantity of air passing in an 
airway and the pressure or water gauge that is produced by the 
circulation. This term refers to the flow of a fluid through an 
orifice in a thin plate, under a given head. The formula 
expressing the velocity of flow through such an orifice is v 

= V2 gh\ multiplying both members of this equation by A, and 
substituting for the first member Av, its value q, after trans¬ 


posing and correcting A — - - - , in which .62 is the coefficient 

.62 >/2 gh 

for the contracted vein of the flow. Reducing this to cubic feet 
per minute and inches of water gauge represented by i, gives 


the equation A = .0004 X 




By this formula, Murgue has 


suggested assimilating the flow of air through a mine to the flow 
of a fluid through a thin plate, for in each case, the quantity 
and the head or pressure vary in the same ratio. Thus, apply¬ 
ing this formula to a mine, Murgue multiplies the ratio of the 
quantity of air passing, in cubic feet per minute, and the square 
root of the water gauge, in inches, by .0004, and obtains an 
area A , which he calls the equivalent orifice of the mine. 

Potential Factor of a Mine. —Ventilating formulas 8 and 27 
page (267) give, respectively, the pressure and the power that will 
circulate a given quantity of air per minute in a given airway. 
These formulas may be written as equal ratios, expressed in fac- 

p ks 

tors of the current and the airway, respectively; thus, — = —, and 

q 1 a 3 

u ks 

— = —■, which show that the ratio between the pressure and the 
q 3 a 3 

square of the quantity it circulates in any given airway is equal 



MINING 


265 


to the ratio between the power and the cube of the quantity it 
circulates. Solving each of these formulas with respect to q: 
With respect to pressure, 



With respect to power, 



Hence, in any airway, for a constant pressure, the quantity 
of air in circulation is proportional to the expression a 

and for a constant power, the quantity is proportional to the 
a 

expression —=, which terms are called the potentials of the 
llks 

mine with respect to pressure and power, respectively; and their 
q q 

values —— and —= are the potentials of the current with respect 
<p \lu 

to pressure and power, respectively. These factors evaluate 
the airway, as they determine the quantity of air a given pres¬ 
sure or power will circulate in that airway, in cubic feet per 
minute. By their use, the relative quantities of air any given 
pressure or power will circulate in different airways are readily 
determined. The rule may be stated as follows: 

Rule .—For any given pressure or power, the quantity of air in 
circulation is always proportional to the potential for pressure, or 
the potential for power, as the case may be. 

This rule finds important application in splitting. In all 
cases where the potential is used as a ratio, the relative potential 
may be employed by omitting the factor k\ or it may be 
employed to obtain the pressure and power, in several splits 
by multiplying the final result by k, as in the splitting form¬ 
ulas 46 and 47, page 275. 

The accompanying table will illustrate the use of the formulas 
used in these calculations. There are several formulas for 
quantity, velocity, and work or horsepower, but in each case 




266 


MINING 


the several formulas are derived by simple transposition of the 

ksv 2 

terms of the original formulas p = -, q — av, and u = qp. 

a 

To illustrate the use of the formulas, take an underground 
road, 5 ft. wide by 4 ft. high, and 2,000 ft. in length, and calcu¬ 
late the value of each symbol or letter, assuming a velocity of 
500 ft. per min. 



Symbol 

Value of 
Symbol 

Area of airway (5 ft.X4 ft.). 

a 

2C sq. ft. 

Horsepower*. 

h 

2.959 H. P. 

Coefficient of frictionf. 

k 

.0000000217 lb 

Length of airway. 

l 

2,000 ft. 

Perimeter of airway, 2X (5 ft.+ 4 ft.) 

o 

18 ft. 

Pressure, in pounds per sq. ft. 

P 

9.765 lb. 

Quantity of air, in cubic feet per 
minute. 

q 

10,000 cu. ft. 

Area of rubbing surface. 

S 

36,000 sq. ft. 

Units of work per minute, power. .. . 

u 

97,650 ft.-lb. 

Velocity, in feet per minute. 

V 

500 ft. 

Water gauge. 

i 

1.87788 in. 

Equivalent orifice of the mine. 

A 

2,919 sq. ft. 

Potential for power. 

Xu 

217.16 units 

Potential for pressure. 

X P 

3,200 units 

Weight of 1 cu. ft. of downcast air. . 

w 

.08098 lb. 

Motive column, downcast air. 

M 

120.5 ft. 

Depth of furnace shaft. 

D 

306.77 ft. 

Average temperature of the upcast 
column. 

T 

350° F. 

Average temperature of the down¬ 
cast column. 

t 

32° F. 


Note. —The water gauge is calculated to 5 decimal places 
to enable all the other values to be accurately arrived at; in 
practice, it is only read to 1 decimal place. 

*A horsepower is equal to 33,000 units of work. 

t This coefficient of friction is an invariable quantity, and is 
the same in every calculation relating to the friction of air 
in mines. 

























MINING 


267 


VENTILATION FORMULAS 


To Find 

No. 

Formula 

To Find 

! No. 

Formula 

Rubbing 






surface of 

1 

s = lo 

gauge in 

14 

• P 

i = ~ 

an airway 
in sq. ft. 



inches. 


o.2 

Area of 


q 

Resistance 



an airway, 
in sq. ft. 

2 

a = 

V 

of an air¬ 
way in lb. 

15 

pa = ksv- 

Velocity 




16 

It 

pa = - 


q 



V 

in ft. per 

3 

v — - 





Q, 




min. 



Quantity 





'X 1 u 

in cu. ft. 

17 

q — av 


4 

' ” \lks 

per min. 


u 


5 



18 


\ks 

u 


19 

a 

X 

| S | <o 

ii 


6 

v — —- 





pa 



3 / 





20 

q= \ “Xa 






\ks 

Pressure 


ksv 2 




in lb. per 
sq. ft. 

7 



21 

q = XuSu 


8 

F a 3 


22 

q = "V.Y phi 





23 

q = Xp^p 


9 

p = - 





*3 

Units of 

24 

u — avp 



10 

p = Mw 

p = 5.2 i 

work per 
minute, 

25 

s 

II 

•Cl 


11 

or power 
on air, in 

26 

a = ksv 3 



12 

j, Q 2 

P Xu 3 

ft.-lb. per 
min. 

27 

ksq 3 

u = -r- 

a 3 


13 

i ** 

II 

■o. 


28 

w = /i33,000 







































268 


MINING 


Table —( Continued ') 


To Find 

No. 

Formula 

To Find 

No. 

Formula 

Units of 
work per 

29 

it 

s 

Pressure 

potential, 

35 


minute, or 


in units 


power on 

30 

tf 3 




air, in ft.- 
lb. per min. 

m = tt- 

Xpi 


36 

X 

II 

•£ih 

Horse- 


U 

Equiva- 


.0004^ 

31 


lent orifice, 

37 


h 


power 

33,000 

in sq. ft. 

Vi 

Power 
potential, 
in units 



M o t i ve 


T — t 

32 

a 

Xu =_ 3rr— 

Xks 

column, 
downcast 
air, in ft. 

38 

M=DX 459+r 





39 

M = - 



3 



w 


33 

S.I* 

II 

a 

Mot ive 
column, 
upcast air, 

40 

T — t 

M - DX 45. J + ( 


34 

s 

II 

in ft. 

39 

fe: 

ii 


As specimen calculations, take a formula for pressure and one 
for quantity. Taking the values from the table and formula 7, 

ksv- 


p =- for pressure 

a 


3 / U 

and formula 20, q= \ — X a, gives— 

\ ks 


.0000000217 X 36,000 X 5002 
p = --—-= 9.765 lb. 


20 


and 


-4 


97,650 


0000000217X36,000 


X 20 =10,000 cu. ft. 


In this way any of these formulas may be worked out. 

Variation of Elements. —In the foregoing table, fixed condi¬ 
tions of motive column, as well as fixed conditions in the mine 
airways, were assumed. It is often convenient, however, to 


































MINING 


269 


know how the different elements, as velocity v, quantity q, 
pressure p, power u, etc., will vary in different circulations; as, 
by this means it is possible to compare the circulations in 
different airways, or the results obtained by applying different 
pressures and powers to the same airway. These laws of 
variation must always be applied with great care. For example, 
before it is possible to ascertain how the quantity in circulation 
will vary in different airways, it is necessary to know whether 
the pressure or the power is constant or the same for each 
airway. The following rules may always be applied: 


For a constant pressure: 


v varies as 



Q 


varies as a 



(relative potential for pressure). 

1 a 

For a constant power: v varies as q varies as —— (relative 

potential for power). [° 

to 

For a constant velocity: q varies as a; p varies as —; u varies 

7 a 

as lo. 

For a constant quantity: v varies inversely as a; p varies 
inversely as X u 3 (potential for power); u varies inversely as 
X u 3 (potential for power) or directly as p. 

For the same airway: The following terms vary as each other: 

v, q, V p , Vtt. 

Similar Airways: r = length of similar side, or similar 
dimension. L 

For a constant pressure: v varies as \/-; q varies as r- 


X r vaf i es as ^ 2 > or ^<7 2 - 


1 


For a constant power: v varies as 3 varies as rX 


4 


r varies as —, or 
Iv 3 


€c 


i 


For a constant velocity: q varies as r 2 ; p varies as u varies 
, r 

i-l u 

as Ir; r varies as \q, —, or —. 

P l 

For a constant quantity: v varies inversely as r 2 ; p and u vary 
r 6 

inversely as —; r varies as 

l 





270 


MINING 


Furnace Ventilation.—p (motive column) varies as D; q varies 
as Vd. 

Fan Ventilation. —It has been customary in calculations per¬ 
taining to the yield of centrifugal ventilators to assume as 
follows: q varies as n\ p varies as n 2 ; u varies as n 3 . 

More recent investigation, however, shows that doubling the 
speed does not double the quantity of air in circulation; or, in 
other words, the quantity does not vary exactly as the number 
of revolutions of the fan. Investigation also shows that the 
efficiency of centrifugal ventilators decreases as the speed 
increases. To what extent this is the case has not been thor¬ 
oughly established. The variation between the speed of a fan 
and the quantity, pressure, power, and efficiency, as calculated 
from a large number of reliable fan tests, may be stated as 
follows: 

For the same fan, discharging against a constant potential: 
q varies as n - 97 ; p varies as n 1-94 . Complement of efficiency 
(1 — K ) varies as n • 425 . 

The efficiency here referred to is the mechanical efficiency, or 
the ratio between the effective work qp and the theoretical work 
of the fan. 

DISTRIBUTION OF AIR IN MINE VENTILATION 

Legal Requirements.—The Anthracite Mine Law of Penn¬ 
sylvania specifies that every mine employing more than 75 
persons must be divided into two or more ventilating districts, 
thus limiting the number that are allowed to work on one air- 
current to 75 persons. The Bituminous Mine Law of Pennsyl¬ 
vania limits the number allowed to work upon one current to 
65 persons, except in special cases, where this number may be 
increased to 100 persons at the discretion of the mine inspector. 

Splitting of Air-Current.—When the air-current is divided 
into two or more branches, it is said to be split. The current 
may be divided one or more times; when split or divided once, 
the current is said to be traveling in two splits, each branch 
being termed a split. The number of splits in which a current 
is made to travel is understood as the number of separate cur¬ 
rents in the mine, and not as the number of divisions of the 
current. 


MINING 


271 


When the main air-current is divided into two or more splits, 
each of these is called a primary split. Secondary splits are the 
divisions of a primary split. Tertiary splits result from the 
division of a secondary split. 

Equal Splits of Air.—When a mine is spoken of as having 
two or more equal splits, it is understood to mean that the 
length and the size of the separate airways forming those splits 
are equal in each case. It follows, of course, from this that the 
ventilating current traveling in each split will be the same, 
inasmuch as they are all subject to the same ventilating pres¬ 
sure, When an equal circulation is obtained in two or more 
splits by the use of regulators, these splits cannot be spoken of 
as equal splits. 

Natural Division of Air-Current.—By natural division of air 
is meant any division of the air that is accomplished without 
the use of regulators; or, in other words, such division of the 
air-current as results from natural means. If the main air- 
current at any given point in a mine is free to traverse two 
separate airways in passing to the foot of the upcast shaft, and 
each of these airways is free or an open split, i. e., contains no 
regulator, the division of the air will be a natural division. In 
such a case, the larger quantity of air will always traverse 
the shorter split of airway. In other words, an air-current 
always seeks the shortest way out of a mine. A comparatively 
small current, however, will always traverse the long split 
or airway. 

It is always assumed, in the calculation of the splitting of 
air-currents, that the pressure at the mouth of each split, 
starting from any given point, is the same. In order to find the 
quantity of air passing in each of several splits starting from a 
common point, the following rule may be applied. 

Rule .—The ratio between the quantity of air passing in any 
split and the pressure potential of that split is the same for all splits 
starting from a common point. Also, the ratio between the entire 
quantity of air in circulation in the several splits and the sum of 
the pressure potentials of those splits is the same as the above ratio, 
and is equal to the square root of the pressure. 

Stated as a formula, indicating the sum of the pressure 
potentials (Ai + Ao+etc.) by the expression 'l.Xp , 


272 


MINING 


2.X P Xi 

Q2 Q3 

Hence, p =-and u = - express the pressure and 

&x P y (xx fi y 

power respectively, absorbed by the circulation of the splits. 
These are the basal formulas for splitting, from which any 
of the factors may be calculated by transposition. 

Proportional Division of Air-Current.—Different proportions 
of air are required in the several splits of a mine than would 
be obtained by the natural division of the air-current. For 
example, the longer splits employ a larger number of men and 
require a larger quantity of air to pass through them. They, 
moreover, liberate a larger quantity of mine gases, for which 
they require a larger quantity of air than is passing in the 
smaller splits. The natural division of the air-current would 
give to these longer splits less air, and to the shorter ones a 
larger amount of air, which is directly the reverse of what is 
needed. On this account, recourse must be had to some means 
of dividing this air proportionately, as required. This is 
accomplished by the use of regulators, of which there are two 
general types, the box regulator and the door regulator. 

The box regulator is simply an obstruction placed in those 
airways that would naturally take more air than the amount 
required. It consists of a brattice or door placed in the entry, 
and having a small shutter that can be opened a certain amount. 
The shutter is so arranged as to allow the passage of more or 
less air, according to the requirements. 

The door regulator divides the air made at the mouth of the 
split. It consists of a door hung from a point of the rib between 
two entries, and swung into the current so as to cut the air like 
a knife. The door is provided with a set lock, so that it may 
be secured in any position,' to give more air to the one or the 
other of the splits, as required. The position of this regulator 
door, as well as the position of the shutter in the box regulator, 
is always ascertained practically by trial. The door is set so as 
to divide the area of the airway proportionate to the work 
absorbed in the respective splits. The pressure in any split 
is not increased, each split retaining its natural pressure. 





MINING 


273 


Calculation of Pressure for Box Regulators.—When any 
required division of the air-current is to be obtained by the use 
of box regulators, these are placed in all the splits, save one. 
This split is called the open, or free, split, and its pressure is 

ksq 1 

calculated in the usual way by the formula p — -. The 

a 3 

natural pressure in this open split determines the pressure of the 
entire mine, as all the splits are subject to the same pressure in 
this form of splitting. 

First, determine in which splits regulators will have to be 
placed, in order to accomplish the required division of the air. 
Calculate the natural pressure, or pressure due to the circulation 
of the air-current, for each split, when passing its required 

ksq 2 

amount of air, using the formula p = -. The split showing 

a 3 

the greatest natural pressure is taken as the free split. In each 
of the other splits, box regulators must be placed, to increase 
the pressure in those splits; or, in other words, to increase the 
resistance of those splits per unit of area. 

The size of opening in a box regulator is calculated by the 
formula for determining the flow of air through an orifice in a 
thin plate under a certain head or pressure. The difference in 
pressure between the two sides of a box regulator is the pressure 
establishing the flow through the opening, which corresponds 

to the head h in the formula v— V 2 gh. This regulator is 
usually placed at the end of a split or airway, and as the regula¬ 
tor increases the pressure in the lesser split so as to make it 
equal to the pressure in the other split, the pressure due to the 
regulator will be equal to the ventilating pressure at the mouth 
of the split, less the natural pressure or the pressure due to 
friction in this split. Hence, when the position of the regulator 
is at the end of the split, the pressure due to friction in the split 

kSQ^" 

is first calculated by the formula p = -, and this pressure is 

a 3 

deducted from the ventilating pressure of the free or open split, 
which gives the pressure due to the regulator. This is then 
reduced to inches of water gauge, and substituted for i in the 








274 


MINING 




formula A = ■ ' ■■ ■The value of A thus obtained is the area, 

V* 

in square feet, of the opening in the regulator. 

By the use of the box regulator, the pressure in all the splits 
is made equal to the greatest natural pressure in any one. 
This split is made the open or free split, and its natural pressure 
becomes the pressure for all the splits, or the mine pressure. 
This mine pressure, multiplied by the total quantity of air in 
circulation (the sum of the quantities passing in the several 
splits), and divided by 33,000, gives the horsepower upon the 
air, or the horsepower of the circulation. 

Size of Opening for a Door Regulator. —The sectional area 
at the regulator is divided proportionately to the work to be 
performed in the respective splits according to the proportion 
Ai: A<i=ui: m. Or as Ai-\-Ai = a, An a = ui : M1 + U2, and Ai 

= —— ~Xa. This furnishes a method of proportionate split- 

K1 + M2 

ting in which each split is ventilated under its own natural 
pressure. The same result would be obtained by the placing 
of the box regulator at the intake of any split, thereby regulat¬ 
ing the amount of air passing into that split, but the door 
regulator presents less resistance to the flow of the air-current. 
The practical difference between these two forms of regulators 
is that in the use of the box regulator each split is ventilated 
under a pressure equal to the natural pressure of the open or 
free split, which very largely increases the horsepower required 
for the ventilation of the mine; while in the use of the door 
regulator each split is ventilated under its own natural pressure, 
and the proportionate division of the air is accomplished with¬ 
out any increase of horsepower. 

In the use of the door regulator, each split is ventilated under 
its own natural pressure, and hence, in the calculation of the 
horsepower of such a circulation, the power of each split must 
be calculated separately, and the sum of these several powers 
will be the entire power of the circulation. 





MINING 


275 


SPLITTING FORMULAS 

Primary Splits.—In the accompanying table are given the 
formulas used in the calculation of primary splits in the natural 
division. The letters represent the same quantities they did 
in the ventilating formulas already given. 


VENTILATING FORMULAS FOR PRIMARY SPLITS 


To Find 

No. 

Formula 

Pressure, potential 

35 

x '-y£ 

^•Xp= (JYi+iY 2 -{-etc.) 

Natural division 

41 


Pressure 

42 

. Q 2 
p (2 a:,)« 

Power 

43 

Q 3 

U (2^ 

Quantity 

44 

45 

Q = S,X p 'fp 

Q= ^(2 X p r-u 

Increase of quantity due to 
splitting the pressure be¬ 
ing constant 

46 

-A-p o 

Increase in quantity due to 
splitting, the power being 
constant 

47 



Example. —What is tjie pressure potential necessary to 
ventilate a mine that has one primary split 4 ft. X 5 ft. X 800 ft. 
and one 4 ft. X 5 ft. X 1.200 ft? 


































276 MINING 

Solution. —Substituting in formula 35, the pressure potential 

of the first split is 20 * -—-=5,060; and for 

\ .0000000217X14,400 

the second split, 20-*/ -—-= 4,131. There- 

\.0000000217X21,600 

fore the pressure potential for the mine is 5,060+4,131 = 9,191 
units. 


VENTILATING FORMULA FOR SECONDARY SPLITS 


To Find 

No. 

Formula 

Relative potential 
for pressure 

35 

X f ,a^ 

Natural division 

48 

41 

Qi = Q - 32 

Q 

3 2 — 

1 + Xl \X 2 2 + (X 3 + X 4 )2 

Pressure 

49 

h k/ Q \ 2/ 

v; v r • i 

\ \X 2 2 ' (X 3 + X 4 ) 2 / 

Power 

50 

. Q z 

u k /, 1 \ 2 

( ' V -+ 1 -) 

\ \AV 1 (X 3 +X 4 ) 2 / 

Quantity 

51 

■ H 

\ \aV^(A 3 +A4) 2 / 












































MINING 


277 


Secondary Splits. —In the calculation of secondary splits 
in the natural division, the work is often shortened, when many 
splits are concerned, by using the relative potential, omitting 
the factor k. But the final result must then be multiplied by k 
to obtain the pressure or power; or, these factors must be 
divided by k, when finding the quantity, as in formulas 49 to 51. 

Example. —How much air, in cubic feet per minute, is 
required to ventilate each split, in the natural division, having 
four secondary splits, as follows: one 4 ft.X5 ft.X800 ft.; 
one 4 ft.X5 ft.X500 ft.; one, 4 ft.X 5 ft.X400 ft.; and one 
4 ft. X 5 ft. X 300 ft? 


Solution. —Substituting in formula 48, the amount of air 
required in the first split is 10,000 — 5,388 = 4,612 cu. ft.; and in 
the second split 


10,000 


1 + .7471 


\l — 1 ~ 

\ 9,42 


: = 5,388 cu. ft. 


,428 2 ' (1.0541 + 1.2172) 2 
The amount of air required in the third split, from formula 41, 


is 


1.0541 


(1.0541 + 1.2172) 
and in the fourth split, 
1.2172 


X5,388 = 2,500 cu. ft.; 


■X 5,388 = 2,888 cu. ft. 


(1.0541 + 1.2172) 

Proportionate Division of Air. —To accomplish the propor¬ 
tionate division of air in primary splits, the pressure in one split 
must be increased by means of a regulator to make it equal to 
the pressure in the free or open split. Hence the pressure due 
to the regulator is equal to the difference between the natural 
pressures in these splits. The pressure due to the regulator 
may, therefore, be found by the formula 

p = p2—pi, 

in which p = pressure due to regulator; 
p 2 = pressure in free split; 

Pi = pressure in regulator split. 

The pressure due to friction may be found by means of for¬ 
mula 13 and the area of the opening in the regulator by formula 
37. 









278 


MINING 


Example. —What is the pressure due to friction if one split 
is 4 ft. X 5 ft. X 800 ft. and has 3,500 cu. ft. per min. of air passing 
through it and the other split is 4 ft. X5 ft. X 1,200 ft. and has 

6.500 cu. ft. per min. of air passing through it? 

Solution. —Substituting in formula 13, the pressure due to 
friction in the first split is 3,500 2 4-5,060 2 = .47845 lb.; and in 
the second split, 6,500 2 4-4,131 2 = 2.4757 lb. 

When using the relative potential, in all calculations relating 
to secondary splits, multiply the result by k to obtain the pres¬ 
sure or the power. To find the pressure due to friction, free 
split, and secondary pressure, use formula 13; the areas of the 
openings in the regulators may be found by formula 37. 

Example. —What is the pressure in each secondary split if 

3.500 cu. ft. per min. of air passes through a split 4 ft.X5 ft. 
X800 ft.; 6,500 cu. ft. per min. passes through a split 4 ft. 
X5 ft.X500 ft.; 4,000 cu. ft. per min. passes through a split 
4 ft.X5 ft.X400 ft.; and 2,500 cu. ft. per min. passes through 
a split 4 ft.X 5 ft.X 300 ft., the constant k in each case being 
.0000000217? 

Solution. —Applying formula 13 and multiplying by k gives 
as the pressure 

In the first split, 

.0000000217 X (3,5004-.7471) 2 = .47848 lb. 

In the second split, 

.0000000217X (6,5004- .9428) 2 = 1.0314 lb. 

In the third split, 

.0000000217 X (4,0004-1.0541) 2 = .31248 lb. 

In the fourth split, 

.0000000217 X (2,5004-1.2172) 2 = .091546 lb. 

As the natural pressure in the third split is greater than that 
in the fourth; the third is the free split and its natural pressure 
is the pressure for the secondary splits. The pressure for the 
primary splits is then found by first adding the pressures in 
the second and third, and if their sum is greater than the natural 
pressure for the first, it becomes the pressure for the primary 
splits, or the mine pressure. If the natural pressure for the 
first is the greater, this is made the free split, and its natural 
pressure becomes the primary or mine pressure. In this case, 


MINING 


279 


the secondary pressure must be increased by placing a regulator 
in the third split. 

If the primary or mine pressure is p 2 -\-pi, the pressure due to 
the regulators is pz — p\, and (P 2 +P 3 ) — pi. 

VENTILATING METHODS AND APPLIANCES 

There are, in general, three systems of ventilation, with 
respect to the ventilating motor employed: natural ventila¬ 
tion, furnace ventilation, and mechanical ventilation. 

Natural ventilation means such ventilation as is secured by 
natural means, or without the intervention of artificial appli¬ 
ances, such as the furnace, or any mechanical appliances by 
which the circulation of air is maintained. In natural ventila¬ 
tion, the ventilating motor or air motor is the air column that 
exists in the downcast shaft by virtue of the greater weight of 
the downcast air and forces the air through the airways of the 
mine. An air column always exists where the intake and 
return currents of air pass through a certain vertical height, 
and have different temperatures. 

In any ventilation, air columns are always established in 
slopes and shafts, owing to the relative temperatures of the 
outside and inside air. The temperature of the upcast, or 
return column, may always be assumed to be the same as that 
of the inside air. The temperature of the downcast, or intake 
column, generally approximates the temperature of the outside 
air, although, in deep shafts or long slopes, this temperature 
may change considerably before the bottom of the shaft or 
slope is reached, and consequently the average temperature 
of the downcast, or intake, is often different from that of the 
outside air. The difference of temperatures will also vary with 
the season of the year. In winter the outside temperature is 
below that of the mine, and the circulation in shafts and slopes 
is assisted, as the return columns are warmer and lighter than 
the intake columns for the same circulation. In the summer 
season, however, the reverse of this is the case. The course 
of the air-current will thus often be changed. When the out¬ 
side temperature approaches the average temperature of the 
mine, there will be no ventilation, except such as is caused by 
accidental wind pressure. 


280 


MINING 


In furnace ventilation the temperature of the upcast column 
is increased above that of the downcast column by means of a 
furnace. The chief points to be considered are the arrangement 
and size of the furnace. Furnace ventilation should not be 
applied to gaseous seams, and in some cases is prohibited by 
law; it is, however, in use in many mines liberating gas. In 
such cases the furnace fire is fed by a current of air taken 
directly from the air-course, sufficient to maintain the fire, 
and the return current from the mines is conducted by means 
of a dumb drift, or an inclined passageway, into the shaft, at a 
point from 50 to 100 ft. above the seam. At this point, the 
heat of the furnace gases is not sufficient for the ignition of the 
mine gases. The presence of carbon dioxide in the furnace 
gases also renders the mine gases inexplosive. In other cases 
where the dumb drift is not used, a sufficient amount of fresh 
air is allowed to pass into the return current to insure its dilu¬ 
tion below the explosive point before it reaches [the furnace. 

In a slope opening, the air column 
is inclined; it is none the less, how¬ 
ever, an air column, and must be 
calculated in the same manner as a 
vertical column whose vertical 
height corresponds to the amount 
of dip of the slope. Fig. 1 shows a 
vertical shaft and a slope, the air 
column in each of these being the same for the same tempera¬ 
ture. The air column in all dips and rises must be estimated 
in like manner, by ascertaining the vertical [height of the dip. 

Calculation of Ventilating Pressure in Furnace Ventilation. 
The ventilating pressure in the mine airways, in natural or in 
furnace ventilation, is caused by the difference of the weights 
of the primary and secondary columns. Air always moves 
from a point of higher pressure toward a point of lower pres¬ 
sure, and this movement of the air is caused by the difference 
between these two pressures. In this calculation each column 
is supposed to have an area of base of 1 sq. ft. Hence, if the 
weight of 1 cu. ft. of air at a given barometric pressure, and 
having a temperature equal to the average temperature of the 
column, is multiplied by the vertical height D of the column, 



B C 

Fig. 1 






MINING 


281 


not only is the weight of the column obtained but the pressure 

at its base due to its weight. Now, as the ventilating pressure 

per square foot in the airway is equal to the difference of the 

weights of the primary and secondary columns, 

/1.3253 X B 1.3253 XM 

p= (--) XD 

\ 459+* 459 + T } 

Calculation of Motive Column or Air Column.—It is often 

convenient to express the ventilating pressure p pound per 
square foot in terms of air column or motive column M, in feet. 
The height of the air column M is equal to the pressure p 


divided by the weight w of 1 cu. ft. of air, or M = —. The expres- 

w 

sion for motive column may be written in terms of the upcast 
air or of the downcast air, the former giving a higher motive 
column than the latter for the same pressure, because the 
upcast air is lighter than the downcast. As the surplus weight 
of the downcast column of air produces the ventilating pressure, 
it is preferable to write the air column in terms of the downcast 
air, or, in other words, to consider the air column as being 
located in the downcast shaft, and pressing the air downwards, 
and through the airways of the mine. If the expression for the 
ventilating pressure is divided by the weight of 1 cu. ft. of 


downcast air 


( L 


3253 X S' 
459+* ) 


after simplifying the motive 


/ T-t \ 

column, M = (-) X D, which is the expression for motive 

\459+77 

column in terms of the downcast air. 

If, on the other hand, the expression for the ventilating pres¬ 
sure is divided by the weight of 1 cu. ft. of upcast air 


-) XD, which is the expression for 

,459+*/ 

motive column in terms of the upcast air. 

Mechanical Ventilators.—A large number of mechanical 
ventilators have been invented and applied to the ventilation 
of mines. Mechanical motors of the fan type present two 
distinct modes of action in producing an air-current: (1) by 
propulsion of the air; and (2) by establishing a pressure due 
to the centrifugal force incident to the revolution of the fan. 







282 


MINING 


Fans have been constructed to act wholly on one or the other 
of these principles, while others have been constructed to act 
on both of these principles combined. 

The action of the disk fan resembles that of a windmill, 
except that in the latter the wind drives the mill, while in the 
former the fan propels the air or produces the wind. This type 
of fan consists of a number of vanes radiating from a central 
shaft, and inclined to the plane of revolution. The fan is set 
up in the passageway between the outer air and the mine air¬ 
ways. Power being applied to the shaft, the revolution of the 
vanes propels the air, and produces a current in the airways. 
The fan may force the air through, or exhaust the air from, the 
airways, according to the direction of its revolution. This 
type of fan is most efficient under light pressures. It has found 
an extensive application in mining practice, but has been 
replaced to a large degree in the ventilation of extensive mines; 
this type of fan acts wholly by propulsion. 

Centrifugal fans include all fans that act solely on the centrif¬ 
ugal principle, and those that combine the centrifugal and pro¬ 
pulsion principles. The action of the fan depends on the form 
of the fan blades, which are set at right angles to the plane of 
revolution, and not inclined, as in the disk fan just described. 
The blades may, however, be either radial, sometimes spoken 
of as paddle blades, or they may be inclined to the radius either 
forwards in the direction of revolution, or backwards. When 
the blades are radial, the action of the fan is centrifugal only. 
The inclination of the blades backwards from the direction of 
motion gives rise to an action of propulsion, in addition to the 
centrifugal action of the fan. The blades in this position may 
be either straight blades in an inclined position, as in the origi¬ 
nal Guibal fan, or they may be curved backwards in the form 
of a spiral, as in the Schiele and Waddle fans. 

Centrifugal fans may be exhaust fans or force fans or blowers. 
In each, the action of the fan is essentially the same; i. e., to 
create a difference of pressure between its intake or central 
opening, and its discharge at the circumference. The centrif¬ 
ugal force developed by the revolution of the air between the 
blades of the fan causes the air within the fan to crowd toward 
the circumference; as a result, a depression is caused at the 


MINING 


283 - 


center and a compression at the circumference, giving rise to a 
difference of pressure between the intake and the discharge of 
the fan. 

Exhaust Fans. —If the intake opening of the fan is placed 
in connection with the mine airways, and the discharge is. 
open to the atmosphere, the fan will act to create a depression 
in the fan drift leading to the mine, which will cause a flow of 
air through the mine airways and into and through the fan. 
In this case, the fan is exhausting, its position being ahead of 
the current that it produces in the airway. The atmospheric 
pressure at the intake of the mine forces the air or propels the 
current toward the depression in the fan drift caused by the 
fan’s action. 

Force Fans and Blowers. —If the discharge opening of the 
fan is placed in connection with the mine airways, a compression 
will result in the fan drift owing to the fan’s action, and the air 
will flow from this point of compression through the airways of 
the mine, and be discharged into the upcast, and thence into- 
the atmosphere. The ventilating pressure in the case of either 
the exhaust fan or the force fan is equal to the difference of pres¬ 
sure created by the fan’s action. In the former case, when the 
fan is exhausting, the absolute pressure in the fan drift is equal 
to the atmospheric pressure less the ventilating pressure, while 
in the latter case, when a fan is forcing, the absolute pressure 
in the fan drift is equal to the atmospheric pressure increased 
by the ventilating pressure. This gives rise to two distinct 
systems of ventilation, known as the vacuum and the plenum 
system. In the vacuum system, the ventilation of the mine 
is accomplished by creating a depression in the return airway 
of the mine. In the plenum system, the air-current is propelled 
through the mine airways by means of the compression or ven¬ 
tilating pressure created at the intake opening of the mine. 

Position of Fan. —The position of the fan, whether used as 
an exhaust or blower, should be sufficiently removed from the 
fan shaft to avoid damage to the fan in case of explosion in the 
mine. Even in non-gaseous mines, the fan should be located 
a short distance back from the shaft mouth, to avoid damage 
due to settlement. Connection should be made with the fan 
shaft by means of an ample drift, which should be deflected 


284 


MINING 



into the shaft so as to produce as little shock to the current as 
possible. In cases of gaseous seams, explosion doors should be 

provided at the shaft 
mouth. The ventilator 
in every large mine 
should be arranged so 
that it may be converted 
from an exhaust to a 
blow-down fan at short 
notice. This is managed 
% by housing the central 
orifices or intake of the 
fan in such a manner as to connect them directly with the fan 
drift. 

Types of Centrigufal Fans.—The Nasmyth fan, shown in 
Fig. 2 is the original type of fan. It had straight paddle 
blades radiating from the center, and was probably the earliest 
attempt to apply the centrifugal principle to a mine ventilator. 
Although not recognized at the time, the fan embodied some 
of the most essential principles in centrifugal ventilation. 

About 1850, Biram attempted to improve upon the Nasmyth 
ventilator by reducing the depth of blade so that it was but 
one-tenth of the radius. In the Biram ventilator, shown in 
Fig. 3, a large number of straight blades were used but they 
inclined backwards from the direction of motion at a con¬ 
siderable angle. This fan was run 
at a considerable speed, but proved 
very inefficient. It depended more 
on the effort of propulsion given to 
the air than on the centrifugal prin¬ 
ciple, as the depth of the blade was 
as much too small as that of Na¬ 
smyth’s was too great. The intake 
or central opening in this fan was as 
contracted as in the former type. 

In the Waddle ventilator, Fig. 4, 
the inventor attempted to reen¬ 
force the discharge pressure at the circumference against the 
pressure of the atmosphere. The discharge took place all 































MINING 


285 


around the entire circumference of the fan, which was entirely 
opened to the atmosphere. The blades were curved backwards 



Fig. 4 


from the direction of motion in spiral form. The width of the 
blade decreased from the throat toward the circumference, so 
as to present an inverse ratio to the length of radius. Thus, 
the area of passage between the fan blades was maintained 
constant from the throat to the circumference of the fan. 



This is the best type of open-running fan having no peripheral 
casing, and discharging air into the atmosphere all around the 
circumference. 

The Schiele ventilator, shown in Fig. 5, was constructed on the 
same principles as the Waddle ventilator, but differed from it 







































286 


MINING 


by having the discharge made into a spiral chamber surround¬ 
ing the fan and leading to an expanding chimney. 

The Guibal ventilator, shown in Fig. 6, embodied the features 
of the Nasmyth ventilator, with the addition of a casing built 
over the fan to protect its circumference. This casing was, 
however, a tight-fitting casing, and as such, differed very 
materially from the Schiele casing. In the Guibal fan the 
blades were arranged upon a series of parallel bars passing upon 
each side of the center and at some distance from it. By this 
construction, the blades were not radial at their inner edge or 
the throat of the fan. They were curved, however, as they 
approached the circumference of the fan, so as to be normal or 
radial at the circumference. 



The Murphy ventilator, shown in Fig. 7, consists of twin fans 
supported on the same shaft and set a few feet apart. Each fan 
receives its air on one side only, the openings being turned 
toward each other. This ventilator is built with a small 
diameter, and is run at a high speed. The blades are curved 
backwards from the direction of motion. The intake opening 
is considerably enlarged; a spiral casing generally surrounds 
the fan, and in every respect this fan makes an efficient high¬ 
speed motor. It has received considerable favor in the United 
States, where it has been introduced into a large number of 
mines. 

Perhaps no centrifugal ventilator has been as little understood 
in regard to its principle of action as the Capell fan, shown in 

















































MINING 


287 


Fig. 8. The fan is constructed along the lines of the Schiele 
ventilator, but differs from that ventilator in the manner of 
receiving its intake air and delivering the same into the main 
body of the fan. A set of smaller supernumerary blades occupy 
a cylindrical space within the main body of the fan, and are 
inclined to the plane of revolution so as to assist in deflecting 
the entering air through small ports or openings into the main 
body of the fan, where it is revolved and discharged at the 
circumference into a spiral space resembling that surrounding 



Fig. 8 


the Schiele fan. The larger blades of this fan are curved back¬ 
wards as the Schiele blades, but are not tapered toward the 
circumference. The fan is capable of giving a high water gauge, 
and is efficient as a mine ventilator. The space surrounding 
the fan is extended to form an expanding chimney. The 
fan may be used either as an exhaust fan or a blower. 
The best results in the United States have been obtained by 
blowers; in Germany, where this fan is in general use, there are 
no blowers. 

Conducting Air-Currents. —A mine door is used for the pur¬ 
pose of deflecting the air-current from its course in one entry 
so as to cause it to traverse another entry, at the same time 
permitting the passage of mine cars through the first entry. 

















288 


MINING 


Stoppings are used to close break-throughs that have been 
made through two entries, or rooms, for the purpose of main¬ 
taining the circulation as the workings advance; also to close 
or seal off abandoned rooms or working places. Stoppings 
must be air-tight and substantially built. 

An air bridge is constructed for the passage of air across 
another airway. When it crosses over, it is called an overcast; 
when it passes under the airway, it is called an undercast. In 
almost every instance, overcasts are preferable to undercasts. 

An air brattice is any partition erected in an airway for the 
purpose of deflecting the current. A thin board stopping is 
sometimes spoken of as a brattice; but the term applies more 
particularly to a thin board or canvas partition running the 
length of an entry or room and dividing it into two airways, 
so that the air will pass up one side of the partition and return 
on the other side, thus sweeping the face of the heading or 
chamber. Such a temporary brattice is often constructed by 
nailing cloth to upright posts set from 4 to 6 ft. apart along one 
side of the entry a short distance from the rib. 

Curtains are sometimes called canvas doors; they are heavy 
duck, or canvas, hung from the roof of the entry to divide 
the air or deflect a portion of it into another chamber or entry. 
Curtains are thus used very often previous to setting a perma¬ 
nent door frame. They are of much use in longwall work, or 
where there is a continued settlement of the roof, which would 
prevent the construction of a permanent door; also, in tempo¬ 
rary openings where a door is not required. 


HOISTING AND HAULAGE 

HOISTING 

There are two general systems of hoisting in use: (1) Hoist¬ 
ing without attempting to balance the load, when the cage and 
its load are hoisted by an engine and lowered by gravity; (2) 
Hoisting in balance, when the descending cage or a special 
counterbalance assists the engine to hoist the loaded ascending 
cage. Hoisting in balance is usually effected by the use of 



MINING 


289 


double cylindrical drums, flat ropes winding on reels, conical 
drums, the Koepe system, and the Whiting system. 

Double cylindrical drums are widely used. They consist 
essentially of an engine coupled directly or else geared to the 
common axis of the drums. The drums are usually provided 
with friction or positive clutches, and brakes, so that they can 
be run singly if desired, or the load can be lowered by gravity 
and the brake. 

Flat ropes wound on reels are sometimes used either for 
unbalanced hoisting with a single reel or for balanced hoisting 
with a double reel. A flat rope has the advantage of prevent¬ 
ing fleeting, but its first cost, extra weight, wear, and difficulty 
of repairing have prevented its very general adoption. 



A conical drum, Fig. 1, equalizes the load on an engine just 
as a flat rope on a reel. On account of the fleeting of the rope, 
however, the drum must be set at a considerable distance from 
the shaft to prevent the rope leaving the head-sheave. A tail- 
rope gives the most perfect counterbalance, the weight of the 
cage and rope on each side being exactly equal. 

In the Koepe system, Fig. 2, one rope runs over and the other 
under driving sheaves S. A tail-rope R is used, and the head- 
sheaves *, x' are placed vertically and at such an angle to each 
other that their grooves and the groove in the driving sheave 
are in line. As the main driving shaft is short, the engines can 
































290 


MINING 


be placed close together, thus requiring a smaller foundation 
and engine house than for a drum hoist. The objection to the 
system is the liability of the rope to slipping about the driving 
sheave, for which reason a hoisting indicator cannot be depended 
on. The system is also inconvenient for hoisting from different 
levels in the same shaft; besides should the rope break, both 
cages will fall to the bottom. 

The Whiting system, Fig. 3, uses two narrow-grooved drums 
placed tandem instead, of a single-driving sheave as in the 

Koepe system. The 
rope passes from the 
cage A over a head- 
sheave, under the guide 
sheave T and around 
the sheaves M and F 
three times, then out to 
the fleet sheave C, back 
under another guide 
sheave, and up over 
another head-sheave to 
the cage B. The sheave 
M is driven by a motor 
either coupled direct to 
its shaft, or geared. The 
drums F and M are 
coupled together by a 
pair of connecting-rods 
like the drivers of a loco¬ 
motive, so that it is 
possible to utilize all the 
friction of both drums to drive the rope. A tail-rope is not 
depended on to produce more friction, though one is generally 
used as a balance to the loads. 

It is best to incline the follower sheave F from the vertical an 
amount equal in its diameter to the distance between the centers 
of two adjacent grooves, the object being to eliminate chafing 
between the ropes around the drums and to prevent them from 
running off by enabling the rope to run from each groove in one 
drum straight to the proper groove in the other. This throws 



Fig. 2 

























MINING 


291 


the shaft and crankpins out of parallel with those of the main 
drum, but this difficulty is overcome by the connections in the 
ends of the parallel rods. The fleet sheave C is arranged to 
travel backwards and forwards, as shown by the dotted lines, 
in order to change the working length of the rope, whereby 
hoisting can be done from different levels in the shaft. 

Power Used for Hoisting. —The power used for hoisting is 
generally steam for the main hoists. Electricity is, however, 
coming rapidly into use, particularly for smaller hoists and 
underground installations, and for main hoists in locations 



where fuel is expensive and water-power available. Gasoline 
engines are also being used to an increasing degree, particularly 
for smaller hoists and in local installations, and they are said 
to give very satisfactory results. 

Problems in Hoisting. —The most common problems in 
hoisting are the following: 

To Balance Conical Drum .—Having given the diameter of 
one end of a conical drum, to determine the diameter of the 
other end that will equalize the load on the engines, call total 
load at bottom A, Fig. 1, empty cage at top B, loaded cage at 













































292 


MINING 


top C, empty cage plus rope at bottom D, small diameter of 
drum x, and large diameter y; then, 

Ax — By = Cy—Dx 

To Find the Size of Hoisting Engine. —Let 
D — diameter of cylinder; 

P = mean effective steam pressure in cylinders; 
r — ratio of stroke to diameter of cylinder; 
w = work per revolution required to be done; 
then, by making one cylinder capable of doing the work, n 
— number of strokes, « = work per minute in foot-pounds. 

3 W 3 U 

D = 1.97 \ — or D = \ -- 

\l Pr \.7854 Prn 

To Find Actual Horsepower of Engine for Hoisting Any Load 
Out of Shaft at Given Rate of Speed. —To the weight of the loaded 
car add the weight of the rope and cage. This will give the 
gross weight. 

Then, 

gross weight in lb. X speed in ft. per min. 

33,000 

Add § for contingencies, friction, etc. 

The following rules regarding winding engines are given by 
Percy: 

To Find Load That Given Pair of Direct-Acting Engines Will 
Start. —Multiply the area of one cylinder by the average pres¬ 
sure of the steam per square inch in the cylinder, and twice the 
length of the stroke. Divide this by the circumference of the 
drum, and deduct ^ for friction, etc. 

Knowing Load and Diameter of Cylindrical Drum, and Length 
of Stroke, Cut-off and Pressure of Steam at Steam Gauge, to Find 
Area and Diameter of Cylinders of Pair of Direct-Acting Engines. 
Multiply the load by the circumference of the drum, and add § 
for friction, etc. Divide this by the mean average steam pres¬ 
sure, multiplied by twice the length of the stroke. 

To Find Approximate Period of Winding on a Cylindrical 
Drum With Pair of Direct-Acting Engines. —Assume the piston 
to travel at an average velocity of 400 ft. per min., and divide 
this by twice the length of the stroke, and multiply by the cir¬ 
cumference of the drum. This gives the speed of cage in feet 





MINING 


293 


per minute. Divide the depth of shaft by this, and the result 
will be the period of winding. 

Head-Frames. —Head-frames are built of wood or steel. 
They vary in height from 30 to 100 ft., depending on local con¬ 
ditions. The inclined leg of a head-frame should be placed so 
as to take up the resultant strain due to the load hanging down 
the shaft and the pull of an engine. 

Fig. 4 shows the graphic method of determining the direction 
and magnitude of this resultant force. Produce the direction 
of the two portions of the rope leading to the drum and down the 
shaft until they intersect at G, measure off a distance GK to 
scale to represent the load hanging 
down the shaft; similarly, measure 
off GH to the same scale to repre¬ 
sent the pull of the engine, complete 
the parallelogram GHLK\ the direc¬ 
tion of the line GL represents the 
direction of the resultant force, and 
its length represents the amount of 
this force. The inclined leg of the 
head-frame should be placed as 
nearly as possible parallel to this 
resultant line, and should be 
designed to withstand a compressive 
strain equal to this resultant. 

Head-sheaves are made of iron, 
being sometimes entirely cast, or 
else the rim and hub are cast separately and wrought-iron spokes 
are used. The former are cheaper and quite satisfactory, 
but the latter are lighter and stronger, and therefore usually 
better. The diameter of the sheave depends on the diameter 
of the rope, and the table giving this will be found on page 106. 
The groove in the sheave should be wood-lined, to reduce wear 
on the rope. Wrought-iron spokes should be staggered in the 
hub and not placed radially. 

Safety catches usually consist of a pair of toothed cams placed 
on either side of the cages and enclosing the guides. When the 
load is on the hoisting rope, these cams are kept away from the 
guides by suitable springs; but if the rope breaks, the springs 

















294 


MINING 


come into action and throw the catches or dogs so that they 
grip the guides, then the tendency to fall increases the grip on 
the guides. 

Detaching hooks are devices that automatically disconnect 
the rope from the cage in case of overwinding. 


HAULAGE 


The magnitude of modern mines and the practice of loading 
or of treating the coal or ore at a large central station makes the 


underground haulage of the material one of the most important 


problems in connection with mining. A good haulage system 
is now essential to make most mines a commercial success. 

Gravity Planes.—With gravity planes the loaded car or trip 
hauls the empty car up the grade. Two ropes are fastened to a 
drum so that the rope attached to the loaded car unwinds from 
the drum as the car descends, while the rope secured to the 
empty car is wound on the drum and the car thus hauled up the 
plane. The following rule gives suggestions based on practice 
that have been successful: For lengths not exceeding 500 ft., 
the minimum grade for the incline should be 5% when the 


weight of the descending load is 
8,000 lb. and that of the ascend¬ 
ing load 2,800 lb. Or the in¬ 
clination should not be less than 



C 


5|% if the respective descending and ascending loads are one- 
half of those just given. When the length of the-plane is from 
500 to 2,000 ft., the grade should be increased from 5% to 10%, 
according to the loads. A load of 4,000 lb. on a 10% grade 
2,000 ft. long will hoist a weight of 1,400 lb. 

The angle of inertia is that angle or inclination at which a car 
will start to move down the slope or plane. The car, when it 
has once started on this grade, will continue to accelerate its 
speed as it descends the plane AB, in the accompanying illus¬ 
tration. If the angle of inclination is decreased until the plane 
AB occupies the position AC, so that the moving car will con¬ 
tinue to move at a uniform velocity instead of accelerating 
its speed, the angle DC A will be the angle of rolling friction, and 
the tangent of this angle will be the coefficient of rolling friction 
for the car. 



MINING 


295 


The upper portion of a plane is made steeper than the lower 
portion so that the trip may start quickly at the head and after¬ 
wards maintain a uniform velocity. With a good brake to 
control the cars, the uniform grade of a central portion of a 
gravity plane should not fall much below 3°, which corresponds 
practically to a 5\% grade. 

Rope Haulage. —The tail-rope system of haulage uses two 
ropes and a pair of drums on the same shaft. The main rope 
passes from one drum directly to the front of the loaded trip, 
and the tail-rope passes from the other drum to the large sheave 
wheel at the end of the road, and back to the rear of the loaded 
trip. While hauling the loaded trip, the drum on which the 
tail-rope is wound is allowed to turn freely on its journal by 
throwing its clutch out, while the engine turns the other drum. 
When the empty trip is being hauled, the clutch on the main- 
rope drum is thrown out and the one on the tail-rope drum is 
thrown in. The engine then turns the tail-rope drum and 
allows the other one to pay out rope as the trip advances. 

The tail-rope system is suitable for steep, circuitous, and 
undulating roads. The trip can be kept stretched at all points, 
and thus the cars will be prevented from bumping together or 
from being jerked apart as the trip is passing over changes in 
the grade. It is undoubtedly the most satisfactory system of 
rope haulage under the natural conditions of most haulage roads 
in mines, and especially so where but one road is available for 
haulage purposes. 

Calculation of Tension of Haulage Rope. —The tension of 
haulage ropes may be found by the formula, 

T = IF(sin a-\-fj. cos a) -f 

in which T = tension or pull upon rope, in pounds; 

W= weight of loaded trip, in pounds; 
w = weight of rope per linear foot, in pounds; 

1 = length of two ropes; equals 2 times the distance 
from winding drum to tail-sheave, in feet; 
d = vertical drop of rope, in feet; 
a — slope angle of maximum grade. 

Example. —What size of steel wire rope will be required to 
haul a trip of 20 mine cars, the weight of the loaded cars being 
3,000 lb. each, the depth of the shaft 300 ft., and the distance 


296 


MINING 


from the foot of the shaft to the “tail-sheave 900 yd., the maxi¬ 
mum grade in this haulage being 10°, /a = ^Assuming a | in. 
rope, weighing .89 lb. per lin. ft. 

Solution.—F rom the formula just given, 

/ .9848\ / 6,000 \ 

r = 60,000X I .17365+-^- ) + .89X ( 300 +“^“) 

= say 12,300 lb., or somewhat over 6 T. 

Referring to the tables for steel haulage ropes with 6 strands 
of 7 wires each, the breaking strain of a f in. rope, weighing .89 
lb. per lin. ft., is found to be 18.6 T. which will give a factor 
of safety of about 3. However, a |-in. or even a 1-in. rope, 
should be used for a change of ropes would then be required less 
often. Making the necessary corrections for 1-in. rope weigh¬ 
ing 1.58 lb. per lin. ft., T = 12,607 lb. 

The endless-rope system uses an endless rope, which is kept 
running continuously by a pair of drums geared together and 
set tandem. The drums are comparatively narrow and pro¬ 
vided with grooves for the rope to run in. Two drums are 
necessary to get sufficient friction to drive the rope when the 
trip is attached to it. The rope is passed around both drums 
a number of times, depending on the amount of friction desired, 
without completely encircling either. It then passes to a ten¬ 
sion wheel at the rear of the drums and thence to the sheave 
wheel at the far end of the road and back to the drums. To be 
used to best advantage, the grade should be in one direction 
and it should be necessary to haul cars from a number of places 
en route. The cars are attached to the rope by friction grips 
in a manner quite similar to the way in which street cars are 
attached to cable lines. Therefore, any jerking due to the cars 
bumping together or stretching the hitchings will seriously 
injure the rope where the grip takes hold. A double road is an 
essential feature of endless-rope haulage. 

To Determine Friction Pull on an Endless-Rope Haulage. 
Let O = output in pounds per minute; 

v = speed of winding, in feet per minute; 

/ = length of haulage road, in feet; 
c — capacity of mine car, in pounds; 
w\ = weight of mine car, in pounds; 
w = weight of rope, in pounds; 



MINING 


297 


T = load on the rope, in pounds; 
fjL = coefficient of friction. 

^ 10 

Then, — = weight of material in transit; 
v 

(l0\ 

2 1 — I u‘i = weight of moving cars, loaded and empty; 
\vc/ 


2lw = weight of rope; 



And if the coefficient of friction equals 



Inclined Roads. —The calculation of power for inclined 
roads is the same as that just given, except that the work due 
to lifting the coal through a height h must be added to that 
found by the previous formulas. If h equals the elevation due 
to the grade of the incline, the additional work of the engine 
due to hoisting the load from this elevation will be Oh and the 
total work per minute u will be 



Motor Haulage. —Wire-rope haulage is very efficient in 
headings, on heavy grades, and against large loads, but in 
crooked passages it entails great costs for renewals and repairs. 
When the grades do not exceed 5% for short distances and 
average 3% against, or for short distances 8% and 5% average 
in favor of loads, locomotives have been found the most econom¬ 
ical form of haulage. Gathering locomotives are used to take the 
cars from the rooms. They are similar in their general con¬ 
struction to the ordinary traction locomotive but are shorter 
and lower. 

In general, it costs from 6 to 10c. per T. to deliver coal from 
face of workings to shaft, slope, or tipple, where the haul is 1 
mi. and the tracks approximately level; yet there are mines 
that at present haul from parting with the trolley system, the 




298 


MINING 


miner delivering from face of room, making an average round 
trip of 9,000 ft., at a total cost of lc. per T. Since the advent 
of the electric-mining locomotive, there has been a change in 
the mine wagons universally used. Formerly -it was customary 
to find as much as 60 lb. per ton car resistance on the level, 
while at present it is as low as 15 lb. 

Compressed air locomotives are particularly useful in gaseous 
mines, as they improve ventilation and are perfectly safe under 
all conditions. Their great disadvantage is their size. 

For a number of years gasoline motors have been used for 
various purposes on the Pacific coast and in metal mines of the 
west, this development being brought about by the high cost 
of steam generation and, in many cases, scarcity of water in 
arid regions. In shape and appearance the gasoline locomotive 
resembles the electric locomotive. They are usually con¬ 
structed to run on full and on half speed. Each motor is 
equipped -with a carbureter, which properly mixes the air and 
the gasoline in the cylinder. Each locomotive is also equipped 
with an electric igniting device, which operates from a storage 
battery when the motor is starting and thereafter from a 
magneto. In some motors, absorption chambers are used to 
absorb the carbon dioxide generated and to cool the gases. 
These chambers are also a protection against the ignition of 
gas or coal dust when the engine back fires. 

The advantages of the gasoline motor are: No power plant 
is needed to operate them, the power-generating apparatus 
being a part of the motor. No transmission-wire lines or pipe 
lines are needed. Humidification of the air is aided. The 
disadvantages are: The use of gasoline in mines, as a mixture 
of gasoline and air forms an explosive gas. Combustion of 
gasoline extracts oxygen from the air. Carbon dioxide and 
nitrogen are products of combustion, and if combustion is not 
complete carbon monoxide is formed. The gasoline motor 
costs 25 to 50% more than an electric motor of the same power. 
The gasoline motor will not start as large a trip or take an over¬ 
load like an electric motor does. 

The Bituminous Mine Laws of Pennsylvania say, “ No prod¬ 
uct of petroleum or alcohol or any compound that in the 
opinion of the inspector will contaminate the air to such an 


MINING 


299 


extent as to be injurious to the health of the miner, shall be 
used as motive power in any mine. ” Modern gasoline motors 
are eliminating this objection more and more as improvements 
are made. 

Speed, of haulage depends on the system of haulage used and 
on the condition of the haulage road. The law in Pennsylvania 
l provides for a speed of haulage not over 6 mi. per hr., and this 
is the speed at which electric and compressed-air haulages are 
usually calculated and at which loaded trips are usually run. 
Empty trips are usually run at a slightly higher speed. It has 
been found in general practice that the maximum pulling power 
of a mule as well as a locomotive is, approximately, one-fifth of 
its weight, or, in other words, a locomotive will pull as much as 
the same weight of mules will pull, and at a speed about three 
times as great. 


MINE ROADS AND TRACKS 

Underground or mine-car tracks should be solidly laid on 
good sills, resting on the solid floor of the mine. They should 
be well ballasted, and should have good clean gutters on the 
lower side of the entry, so that the rails may be protected as 
much as possible from the action of the mine water. 

The grades depend entirely on circumstances, but, when 
possible, the grade should be in favor of the load, and should 
be at least 5 in. in 100 ft. to insure flow in the gutters alongside 
the track. 

Ties should be spaced about 2 ft. apart, center to center, 
making 15 to a 30-ft. rail. The rail should be well spiked to the 
ties with four spikes to each tie, the joint between two rails on 
one side of the track being located about midway between two 
joints on the opposite rail. On curves, ties should be laid 
so as to form radii of the curves of the track. 

The weight of rail to be chosen in any individual case depends 
entirely on the weight of cars used, and the motive power. For 
cars having a capacity of about 1§ T., the weight of rail, when 
the motive power is live stock, should not be less than 16 lb. 
per yd., while for cars having a capacity of 2 T. or over, a 20-lb. 
rail should be used. There is no economy in using a very light 
rail, as the base is gradually eaten away by the mine water; 


300 


MINING 


a heavy section of rail can be used much longer before the rails 
become weakened. On main roads, where haulage machinery 
of one kind or another is used, the weight of rail for 2-T. cars 
should be from 25 lb. to 35 lb. per yd., and on steep slopes as 
high as 40 lb. per yd. In the case of locomotive haulage, 
authorities claim that the weight of rail should be regulated 
by allowing 1 T. for each driver for each 10 lb. weight of rail 
per yd. 

The gauge of the track in coal mines should not be less than 
30 in. nor more than 48 in. A mean between these two, or a 
gauge of from 38 in. to 42 in. is desirable, because it com¬ 
bines, to a certain extent, the advantages claimed for the 
extremes. 

Curves should be of as large a radius as permissible, and 
never, if possible, of less radius than 25 ft. The resistance of 
curves is considerable; and the smaller the radius of the curve, 
and the greater the length of the curved track occupied by the 
trip, or train, the greater is the resistance. 

To Bend Rails to Proper Arc for Any Radius. —Rails are 
usually 30 ft. long, and the most convenient chord to use in 
bending mine rails is 10 ft. Then, having the radius and chord, 
find the rise of middle ordinate by squaring the radius, and 
from it take the square of one-half the chord. Extract the 
square root of the remainder and subtract it from the radius; 
the result will be the rise of the middle ordinate. Thus, having 
a radius of 30 ft. and a chord of 10 ft. the middle ordinate will be, 

30- V30 2 — 5 2 = .42 ft. 

Rail Elevation. —In elevating rails on curves, consider 
whether the hauling is to be done by a rope, a locomotive, or an 
electric motor. For either of the latter, elevate the rail on the 
outside of the curve; but for the first, elevate the inner rail, 
for as the power is applied by a long flexible rope, there is 
always a tendency for both rope and cars to take the long chord 
of the curve as soon as the point of curve is reached. On slope 
haulages, operated by a single rope, when the weight of the cars 
traveling on the grade of the slope is sufficient to draw the rope 
off the hoisting drum, the rails on curves should be elevated on 
the outside, the effect then being similar to that of a locomotive, 
i. e., the centrifugal force tends to throw the car to the outside 



MINING 


301 


of the track. In such cases, the elevation should be moderate 
so as not to interfere with the trip when drawn out again by the 
rope—the opposite effect being then experienced. 

Rollers. —The rollers on level tracks should not be more 
than about 20 ft. apart to properly carry the rope, and on gravity 
slopes where the lower end of the slope gradually flattens off, 
the distance between rollers should not be more than 12 to 15 
ft., as this spacing allows the trip of cars to run much farther 
by keeping the rope well off the ties, than if they are farther 
apart, thereby not supporting the rope, and causing a great 
amount of friction between the rope and the ties. 

Switches. —The switch, or latch, most commonly used in 
mines is shown in Fig. 1. When the branch or siding is in 
constant use, an ordinary railway frog is substituted for the 
cross-bar b. The latches a, are wedge-shaped bars of iron 
(made as high as the rail) with an eye in the thick end. 

A modification of this 
switch is shown in Fig. 2, 
which represents a form of 
double switch. These 
latches are set by the 
drivers, who kick them over 
and drop a small square of 
plate iron between them to 
hold them in place. This 
switch costs more than the 
other style and is better 
adapted to outside roads than to inside roads. The ordinary 
movable rail switch in every-day use on all surface railways is 
sometimes used in mine roads. It is commonly used in slopes 
arranged as shown by Fig. 6, to replace latches set by the car, 
and is also largely used in outside roads. 

For crossings, ordinary railway frogs and grade crossings are 
sometimes used, as is also a small turntable, which then answers 
two purposes. More frequently the plan shown in Fig. 3, in 
which four movable bars are thrown across the main track when¬ 
ever the other road is to be used, is adopted. The subordinate 
road is built from to 2 in. higher than the main road, to 
allow the bars to clear the main-track rails. 
















302 


MINING 


Turnouts. —On gangways or headings used as main haul¬ 
age roads, turnouts should be constructed at convenient 



intervals to allow the loaded and empty trips to pass. These 
turnouts should be long enough to accommodate from 5 or 6 
up to 15 or 20 cars. 

Slope Bottoms.—At the foot of a slope or at the landing on 
any lift, the gangway is widened to accommodate at least two 
tracks—one for the empty and one for the loaded cars. The 


empty track should be on the 



Fig. 4 


upper side of the gangway, or 
that side nearer the floor of 
the seam, and the loaded track 
on that side of the gangway 
nearer the roof of the seam. 
An arrangement of tracks often 
used is shown in Figs. 4 to 9. 



Material Required for 1,000 Ft. and for 1 Mi. of Single 
Track. —For the quantity of wooden ties required for 1,000 ft. 




































































22EZZ 


MINING 


303 


and for 1 mi. of single track, see page 307. Weights of rails 
are given in long tons of 2,240 lb.; hence .9 T. is equal to 




2,016 lb. and not to 1,800 lb. Each increase of 5 lb. per yd. 
in the weight of the rail, increases by 1.488 T. and by 7.857 T., 
respectively, the tons required to lay 1,000 ft. or 1 mi. of track. 
In measuring a rail, it will be found that the height of a rail is 
equal to the width of its base. 

Rule .—To find the weight, in long tons, of the rails necessary 

to lay 1 mi. of single track, multiply the 
weight per yard of the rail by ty, or by 
1.5714. 

Rule .—To find the weight of rails for 
1,000 ft. of single track, multiply the 
weight per yard by .29761. 

Thus, the weight of 70-lb. steel for 
1 mi. and for 1,000 ft. of single track 
would be, respectively, 70X-¥-= 110 T., 
and 70 X.29761 =20.833 T. 









xx 









Fig. 9 


For lengths other than 1,000 ft., multiply the quantities for 
1,000 ft. by the ratio the given length of track bears to 1,000. 






















































304 


MINING 


Thus for the material for 600 ft., 1,580 ft. or 4,000 ft., multiply 
the quantities (rails, fish-plates, bolts, or spikes) by .6, 1.58 or 
by 4 as may be. 

Prices quoted for rails include the necessary splice bars 
(fish-plates) and bolts, but not the spikes. If requested at the 
time of placing the order, the mills will drill the holes necessary 
for electric bonding, and, generally, without charge. While the 


STANDARD SIZE OF RAILS 


Weight 
per Yard 


Width 
of Head 

Amount 

Amount 

Height 

Required 
per 1,000 

Required 
per Mile 

Pounds 

Inches 

Inches 

ft. Tons 

Tons 

8 

H 

XA 

16 

2.381 

12.571 

12 

11 

ll6 

3.571 

18.857 

16 

21 

U 

4.762 

25.144 

20 

21 

U 

5.952 

31.429 

25 

2f 

H 

7.441 

39.286 

30 

3 

if 

8.929 

47.143 

35 

31 

H 

10.417 

55.000 

40 

31 

H 

11.905 

62.857 

45 

3 H 

31 

2 

13.393 

70.714 

50 

21 

14.881 

78.571 

55 

4A 

1 

4 

16.369 

86.428 

60 

41 

21 

17.858 

94.286 

65 

4tV 

013 

19.346 

102.143 

70 

4! 

OJL. 

•^16 

20.833 

110.000 

75 

4ii 

015 

22.321 

117.857 

80 

5 

21 

23.809 

125.714 

85 


2A 

2f 

25.298 

133.571 

90 

51 

26.786 

141.429 

95 

5A 

2H 

28.274 

149.286 

100 

51 

21 

29.763 

157.143 


standard length of rails is 30 ft., the order is considered to 
have been acceptably filled, if not to exceed 10% of the rails 
are in shorter lengths, varying by even feet down to 24 ft. In 
the accompanying table all sizes from 40 lb. to 1001b., are of 
the standard established by the American Society of Civil 
Engineers. A certain quantity of these standard sizes are 
usually in stock, insuring the prompt filling of small orders. 














MINING 


305 


MATERIALS REQUIRED FOR SINGLE-TRACK ROAD 


Size of 
Road 

Number 
of 30-Ft. 

Rails 

Required 

Number 
of Splices 
Required 

Number of Bolts 
Required 

4 per Joint 

6 per Joint 

1,000 ft. 

1 mi. 

68 

352 

236 

704 

272 

1,408 

408 

2,112 


SIZES AND QUANTITIES OF SPIKES REQUIRED FOR 
TIES 2 FT., CENTER TO CENTER, 4 SPIKES 
PER TIE 


Size 

Measured 

Under 

Head 

Inches 

Average 
Number 
per 
Keg 
of 200 
Lb. 

Quantity Required 

Weight 
of Rail 
Used 

Pounds 

per 1,000 Ft. 
Track 

per Mile of 
Track 

Lb. 

Kegs 

Lb. 

Kegs 

2*Xt 

1,342 

300 

H 

1,575 

71 

8 to 16 

3 XI 

1,240 

324 

if 

1,710 

81 

16 to 20 

31X1 

1,190 

340 

H 

1,780 

9 

16 to 20 

4 XI 

1,000 

360 

if 

2,090 

101 

16 to 25 

31XA 

900 

445 

21 

2,350 

11 

16 to 25 

4 XA 

720 

550 

21 

2,910 

14f 

20 to 30 

4|XtV 

680 

590 

3 

3,110 

151 

20 to 30 

4 XI 

600 

670 

31 

3,520 

171 

25 to 35 

41X1 

530 

750 

31 

3,960 

20 

30 to 35 

5 XI 

450 

880 

4f 

4,660 

231 

35 to 40 

5 XA 

400 

980 

5 

5,170 

26 

40 to 55 

5iXAr 

375 

1,112 

5| 

5,870 

291 

45 to 75 

5§X | 

300 

1,334 

6§ 

7,040 

35f 

75 to 100 


Note. —When ordering spikes, a reasonable allowance should 
be made for waste. For ordinary mine track with 2 spikes to 
the tie, divide the quantities given in the table by 2. For 
other spacing than 2 ft., proceed as follows: For 30 in., 
multiply the quantity of spikes by .8; for 28 in., by .858; 
for 26 in., by .893; for 22 in., by 1.092; for 20 in., by 1.2; and 
for 18 in., by 1.334. 




































NUMBER OF BOARD FEET IN MINE TIES 


306 



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MINING 


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MINING 


307 


TIES PER 1,000 FT. AND PER MILE OF TRACK 


Length 

of 

Track 

• 

Distance From Center to Center of Ties, in Inches 

18 

20 

22 

24 

26 

28 

30 

Number of Ties 

1,000 ft... 
1 mi. 

667 

3,520 

600 

3,168 

546 

2,880 

540 

2,640 

462 

2,437 

429 

2,267 

400 

2,112 


Example. —How many feet, board measure, are there in the 
ties required to lay 1,500 ft. of track; the ties are 6 ft. 6 in. long, 
5 in.X6 in. in cross-section, and spaced 22 in. between centers? 

Solution. —1,500 ft. = 1§ thousands of feet. From the 
accompanying tables, 1§ X546X 16£ = 13,308f, say, 13,500 ft., 
B. M. 


THE PREPARATION OF COAL 

CRUSHING MACHINERY 

The object of crushing ore or coal is: first, to free the min¬ 
eral or other valuable constituents from the gangue, slate, 
pyrites (sulphur), or other worthless or objectionable constit¬ 
uents so that they can be subsequently separated; or, second, 
simply to reduce the size of the individual pieces and so get the 
material into a more salable or convenient condition for use. 

Selection of a Crusher. —The style of crusher employed is 
influenced by: The amount of material to be crushed in a 
given time. The size of the material as it goes to the crusher. 
The physical characteristics of the material to be crushed; 
that is, whether it is hard or soft, tough or brittle, clayey 
or sticky. The object of the crushing. The character of 
the product desired; that is, whether an approximately sized 
product is desirable and whether dust or fine material is 
objectionable. 























308 


MINING 


The term cracking rolls is applied to rolls having teeth, which 
are usually made separate and inserted. These rolls, Fig. 1, 
are used for breaking coal, phosphate rock, etc., the object being 
to break the material into angular pieces with the smallest 
possible production of very fine material. The principle field 
for cracking rolls is in the preparation of anthracite, and the 
exact style or design of the roll depends largely on the physical 
condition of the coal under treatment. In most cases, the rolls 
are constructed with an iron cylinder having steel teeth inserted, 
the size, spacing, and form of the teeth depending on the size 
and physical condition of the material to be broken. Cracking 


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fi\ 

jt--r 

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© © 

T^T 


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rolls vary from 12 to 48 in. in diameter and from 24 to 36 in. in 
surface width. The teeth of the larger sizes are from 3 to 
3J in. high, and of the smaller 1 in. or less. 

The form of the teeth varies greatly, but, as a rule, the larger 
rolls have straight pointed teeth of the sparrow-bill or some 
similar form, Fig. 2 a. The old curved, or hawk-billed, teeth. 
Fig. 2 b, have now gone wholly out of use. 

Disintegrating rolls and pulverizers are sometimes used to 
reduce coking coal to the size of corn or rice before intro¬ 
ducing it into the ovens. One roll is driven at double the 
speed of the other, the slower roll acting as a feed roll, and 
the other as a disintegrator. 









































































MINING 


309 


For the reduction of coal, crushers employing hammers have 
been used, Fig. 3. The crushing chamber is usually of a cir¬ 
cular or barrel form, and the crushing is done by means of 
hammers pivoted about a central shaft. These swing out by 
centrifugal force and strike blows upon the coal to be broken. 



When it is reduced sufficiently fine, it is discharged through 
bars or gratings at the lower portion of the machine. This 
style of machinery is usually employed in preparing coal for 
coke ovens, thus occupying the same field as the disintegrating 
rolls. 


SIZING AND CLASSIFYING APPARATUS 
Platform Bars. —In the anthracite breakers, the terms plat¬ 
form bars or head-bars are usually employed for the bars that 
remove the fine material from the run of mine, so that the 
coarse will go to the crushers. These bars are made of l§-in. 
to 2-in. round iron placed at an inclination of a-in. to 1-ft., the 
spacing depending on the size of coal it is desired to make in the 
breaker. 

Shaking Screens. —Shaking screens have an advantage in 
that the entire area of the screen is available for sizing, and 
hence a greater capacity can be obtained from a given area of 
screening surface. They also occupy less vertical height than 
a revolving screen. In coal breakers they are particularly 










































310 


MINING 


applicable where the coal is wet and has a tendency to stick 
together. The principal disadvantage of the shaking screen 
is that the reciprocating motion imparts a vibration to the 
framing of the building. 

The capacities of shaking screens operating on anthracite 
have been given as follows. The parties giving these figures 
advise the use of 140 R. P. M. for the cam-shaft. For broken 
and egg coal, J sq. ft. per T. for 10 hr. For stove and chestnut 
coal, J sq. ft. per T. for 10 hr. For pea and buckwheat coal 
f sq. ft. per T. for 10 hr. For birdseye and rice, 1? sq. ft. per T. 
for 10 hr. For sizing bituminous coal, inclined shaking screens 
are extensively used in certain sections, particularly in the 
Middle Western States. These screens are given a shaking 
motion by means of cams and connecting-rods, which make 
from 60 to 100 strokes per minute, the speed varying according 
to the amount of moisture in the coal. 

Size of Mesh. —The perforations given in the accompanying 
table have been adopted by two of the largest anthracite 
companies as the dimensions for the holes in shaking screens 
to produce sizes equivalent to those produced by revolving 
screens. 


MESH FOR SHAKING SCREENS 


Kind of Coal 

Lehigh 
Valley 
Coal Co. 

Phila. & Reading 
Coal & Iron Co. 

Kind of Coal 

Round 

Inches 

Round 

Inches 

Square 

Inches 

Steamboat . . . 


5f 

5 

Steamboat 

Lump. 

4i 

4 h 

4 

Large broken 

Broken. 

3* 

31 

2f 

Small broken 

Egg. 

OJL 

•^16 


2 

Egg 

Stove. 

H 

H 

A 8 

Stove 

Chestnut. 

H 

7 

8 

3 

Chestnut 

Pea. 

5 

8 

0 

re 

1 

2 

Pea 

Buckwheat. . . 

11. 

32 

5 

16 

1 

4 

Buckwheat 

Rice. 

. S 

16 

_S_ 

16 

1 

8 

Rice 






















MINING 


311 


Revolving Screens, or Trommels. —The screen is placed 
about the periphery of a cylinder or frustum of a cone. The 
material to be sized is introduced at one end; the small size 
passes through the screen, and the other size is discharged from 
the other end. If the form is cylindrical, the supporting shaft 
must be inclined so that the material will advance toward the 
discharge end. The inclination determines the rapidity with 
which the material will be carried through the screen. The 
advantage of the conical screen is that the shaft is horizontal 
and hence the bearings are simpler; this a very decided advan¬ 
tage where the machinery must be crowded into a minimum 
space, and is hard to get at. 

Speed. —The periphery of a revolving screen should travel 
about 200 ft. per min. In the case of very fine material, 
screens are sometimes run faster than this. The following 
have been adopted as standard speeds for screens by one of the 
largest anthracite companies: 

Speed of Screens 

Rev. per Min. Rev. per Min. 

Mud-screens.8.87 Big screens. 8.52 

Counter mud screens . . 15.49 Pony screens.10.87 

Cast-iron screens.11.25 Buckwheat screens. . . . 15.30 

Duty of Anthracite Screens. —The following list gives the 
number of square feet of screen surface required for a given duty 
in the case of revolving screens working upon anthracite: 


Egg Coal. IT. per 1 sq. ft. per 10 hr. 

Stove coal. IT. per 1* sq. ft. per 10 hr. 

Chestnut coal. IT. per 1? sq. ft. per 10 hr. 

Pea coal. IT. per 2 sq. ft. per 10 hr. 

Buckwheat coal. IT. per 2f sq. ft. per 10 hr. 

Rice coal. IT. per 3f sq. ft. per 10 hr. 

Culm. IT. per 5 sq. ft. per 10 hr. 


These figures may be reduced from 20% to 30% for very 
dry or wash coal. 

Revolving Screen Mesh for Anthracite. —A standard mesh 
for revolving screens for sizing anthracite was adopted some 
years ago, but it is only approximately adhered to and a 
considerable variation from the standard is found throughout 
the anthracite region. The following are probably as nearly 
standard meshes for revolving screens for sizing anthracite 
coal as can be given: 













312 


MINING 


Mesh for Sizing Coal 

Culm.passes through 3 V-in. mesh 

Birdseye .passes over f-in. mesh, and through 

j^-in. mesh 

Buckwheat . . .passes over f-in. mesh, and through 
5 -in. mesh 

Pea.passes over 5 -in. mesh, and through 

f-in. mesh 

Chestnut.passes over f-in. mesh, and through 

lf-in. mesh 

Stove.passes over lf-in. mesh, and through 

2 -in. mesh 

Egg..passes over 2 -in. mesh, and through 

2 f-in. mesh 

Grate.passes over 2f-in. mesh, and out end 

of screen 


Special grate.. passes over 3-in. mesh, and out end 
of screen 

Special steamboat passes over 3-in. bars, and through 
6 -in bars. 

Jigs.—The general term jig is applied to that class of con¬ 
centrating machines in which the separation of the mineral 




Fig. 4 


from the gangue takes place on a screen or bed of material and 
is effected by pulsating up-and-down currents of a fluid medium. 
A number of different methods are used for driving the pistons 
that cause the pulsations of the water in jigs. Some use plain 
eccentrics, giving the same time to both the up and the down 
strokes of the pistons, while others employ special arrangements 
































































































MINING 


313 


of parts, which give a quick down stroke and a slow up stroke, 
thus allowing the water ample time to work its way back 
through the bed without any sucking action from the piston. 
This tends to make a better separation in some cases than the 
use of the plain eccentrics. 

Stationary screen jigs are illustrated by Pig. 4, which shows a 
3-compartment jig. The separation takes place on screens 
supported on wooden frames g, and is effected by moving the 
water in each compartment so that it ascends through the 
screen, lifting the mineral and allowing it to settle again, thus 
giving the material an opportunity to arrange itself according 
to the law of equally falling particles. 

Removal of Sulphur from Coal. —The object of washing coal 
is to remove the slate and pyrites, thus reducing the amount of 
ash and sulphur. Many forms of washers easily and cheaply 
reduce the slate from 20% in the coal to 8% of ash in the coke, 
but it is much more difficult to reduce 4% of sulphur in thg 
coal to 1 % or less of sulphur in the coke. Sulphur occurs in the 
coal in three forms, as hydrogen sulphide, calcium sulphate, 
and pyrite. 

HANDLING OF MATERIAL 

Anthracite.—The following may be taken as average figures 
for the angle or grade of chutes for anthracite, to be used where 
the chutes are lined with sheet steel: For broken or egg coal, 
2\ in. per ft.; for stove or chestnut coal, 3| in. per ft.; for pea 
coal, 4i in. per ft.; for buckwheat coal, 6 in. per ft.; for rice 
coal, 7 in. per ft.; for culm, 8 in. per ft. 

If the coal is to start an the chute, 1 in. per ft. should be 
added to each of the foregoing figures; while if the chutes are 
lined with manganese bronze in place of steel, the figures can be 
reduced 1 in. per ft. for coal in motion, or would remain as in the 
table to start the coal. When the run of mine is to be handled, 
as in the main chute, at the head of the breaker, the angle 
should be not less than 5 in. per ft., or practically 22^° from the 
horizontal. If chutes for hard coal are lined with glass, the 
angle can be reduced from 30% to 50%, depending somewhat 
on the nature of the coal. In all cases, the flatter the coal, the 
steeper the angle must be, on account of the large friction sur¬ 
faces exposed, compared with the weight of the piece. If the 


314 


MINING 


chutes are lined with cast iron, the angle should be about the 
same as that employed for steel, though sometimes a slightly 
greater angle is allowed. 

The accompanying table is printed through the courtesy of 
the Link-Belt Engineering Co., Philadelphia, Pa.: 


PITCH AT WHICH ANTHRACITE WILL RUN 


Size of Coal 

Dry Coal 

Wet Coal 

Sheet- 

Iron 

Lining 

Cast- 

Iron 

Lin¬ 

ing 

Glass Lining 

Pitch of Chute, in Inches per Foot 

Start On 

Continue On 

Start On 

Start On 

Continue On 

l 

Start On 

Continue On 

1 

Broken slate. 

51 

41 

51 

31 

3 



Dry egg slate.... 

51, 

41 

51 

31 

3 



Dry stove slate.. . 

51 

41 

51 

31 

3 



Dry chestnut slate 

5 

41 

51 

31 

3 



Broken coal. 




2f 

21 



Egg coal. 

31 

3 

31 

2! 

21 

21 

If 

Stove coal. 

4f 

41 

41 

3 

21 

21 

21 

Chestnut coal. . . . 

41 

41 

41 

3 

21 

3 

21 

Pea coal. 

51 

5 

51 

31 

21 

3 

2^ 

Buckwheat No. 1. 




31 

31 

31 

31 

Buckwheat No. 2. 




31 

31 

31 

31 

Buckwheat No. 3. 




41 

31 

41 

41 

Buckwheat No. 4. 




41 

41 

41 

41 


Bituminous Coal.—When the run of mine is to be handled, 
the angle of the chutes should be from 35° to 45° from the hori¬ 
zontal, or from 85 in. to 12 in. per ft. If the coal is wet, the 
angle should always be steeper, as coarse coal will slide on a 
flatter angle than slack or fine coal. 































MINING 


315 


BRIQUETING 

Fuel, fuel dust, and other products may be briqueted by a 
number of different styles of machines, but all these may be 
divided into two classes, briquet and eggette machines. Fuel 
briquets have not come into general use in the United States 
on account of the great amount of cheap fuel available, which 
has prevented the utilization of culm, coal dust, etc.; and on 
account of the lack of or high price of suitable bonding material. 
This latter condition is now being removed by the introduction 
of by-product coke ovens, from which supplies of coal tar can be 
obtained. 

SPACE OCCUPIED BY 2,000 LB. OF VARIOUS 

COALS 


Anthracite 

Broken 

Cubic 

Feet 

Egg 

Cubic 

Feet 

Stove 

Cubic 

Feet 

Chestnut 

Cubic 

Feet 

Pea 

Cubic 

Feet 

Lackawanna. 

37.10 

36.65 

34.90 

34.35 

37.25 

Garfield red ash.... 

37.30 

36.95 

36.35 

36.35 

37.50 

Lykens Valley. 

37.55 

37.25 

37.55 

37.25 

38.50 

Shamokin. 

38.05 

37.70 

37.25 

37.25 

38.50 

Plymouth red ash.. . 

34.90 

34.85 

34.75 

34.70 

36.90 

Wilkes-Barre. 

34.95 

34.35 

33.75 

34.00 

36.90 

Lehigh. 

33.30 

33.80 

33.55 

32.55 

33.05 

Lorberry. 

34.65 

34.20 

33.80 

33.55 

35.20 

Scranton. 

35.35 

35.20 

34.60 

33.30 

34.95 

Pittston. 

35.45 

34.95 

34.35 

33.70 

35.50 


Bituminous 

Cubic 

Feet 

Bituminous 

Cubic 

Feet 

r .11 m hprl anH 

36.65 

33.55 

40.15 

Pocahontas. 

34.00 

41.50 

42.30 

Clearfield. 

New River. 

American cannel. 

English cannel. 






































316 


MINING 


TREATMENT OF INJURED PERSONS 

The dangers to be feared in case of wounds, are shock or 
collapse, loss of blood, and unnecessary suffering in the moving 
of the patient. 

In shock, the injured person lies pale, faint, and cold, some¬ 
times insensible, with feeble pulse and superficial breathing. 
The cause of death in case of a shock is arrest of heart action, 
produced by the suspension of the functions of the brain and 
spinal cord. In treatment, the two most important parts are: 
the position of the injured person and the application of external 
warmth. 

The injured person should at once be placed in a recumbent 
position, his head resting on a plane lower than that of his 
trunk, legs, and feet. He should be well wrapped up and pro¬ 
tected from the chilling influences of external air. Where there 
is danger of immediate death, stimulants should be given; in 
all other conditions of shock, stimulants are injurious. 

Loss of Blood.—In case of loss of blood, two conditions 
present themselves: (1) The bleeding is arrested spontaneously 
or otherwise, but the injured person presents all the symptoms 
of loss of blood; (2) the injured person is actually bleeding, 
and he is, or is not, suffering from loss of blood. 

In the first condition, life is threatened by anemia of the 
brain and spinal cord, and all the efforts of treatment should 
be to direct the flow of whatever quantity of blood that may 
still remain in the body to these vital centers. This is most 
efficiently done by placing the injured person in a recum¬ 
bent position, with his head resting on a plane somewhat lower 
than that of his trunk and legs. In graver cases, constricting 
bands should be applied to both arms, as near the shoulders as 
possible, and to both thighs, as near the abdomen as possible. 
This last maneuver directs the entire quantity of blood in the 
body to the suffering centers, the centers of life itself. Stimu¬ 
lants may be sparingly administered. 

If there is bleeding, do not try to stop it by binding up the 
wound. The current of blood to the part must be checked. 
To do this, find the artery, by its beating; lay a firm and even 
compress or pad (made of cloth or rags rolled up, or a round 



MINING 


317 


stone or piece of wood well wrapped) over the artery, as shown 
in Fig. 1. Tie a handkerchief around the limb and compress; 
put a bit of stick through the handkerchief and twist the latter 




up until it is just tight enough to stop the bleeding; then put 
one end of the stick under the handkerchief, to prevent untwist¬ 
ing, as in Fig. 2. 

The artery in the thigh runs along the inner side of the 
muscle in front of the bone, as shown by dotted line in Fig. 3. 
A little above the knee it passes to the back of the bone. In 
injuries at or above the knee, apply the compress higher up, on 
the inner side of the thigh, at the point P, Fig. 3, with the knot 
on the outside of the thigh. When the leg is injured below the 
knee, apply the compress at the back of the thigh, just above 
the knee, at P, Fig. 4, and the knot in front, as in Figs. 1 and 2. 

The artery in the arm runs down the inner side 
of the large muscle in front, quite close to the 

bone, as shown by dotted line; 
low down it is further for¬ 
wards, toward the bend of the 
elbow. It is most easily com¬ 
pressed a little above the mid¬ 
dle, at P, Fig. 5. Care should be 
taken to examine the limb from 
time to time, and to lessen the 
compression if it becomes cold 
or purple; tighten up the hand¬ 
kerchief again if the bleeding 
begins afresh. 

To Transport a Wounded Person Comfortably. —Make a 
soft and even bed for the injured part, of straw, folded blankets, 
quilts, or pillows, laid on a board with side pieces of board 
nailed on, when this can be done. If possible, let the patient 




Fig. 4 








318 


MINING 


be laid on a door, shutter, settee, or some firm support, properly- 
covered. Have sufficient force to lift him steadily, and let those 

that bear him not keep step. 

Should any important ar¬ 
teries be opened, apply the 
handkerchief, as recom¬ 
mended. Secure the vessel by 
a surgeon’s dressing forceps, 
or by a hook, then have a 
silk ligature put around the 
vessel, and tighten. Should the bleeding be from arterial 
vessels of small size, apply persulphate of iron, either in 
tincture or in powder, by wetting a piece of lint or sponge 
with the solution; then, after bleeding ceases, apply a compress 
against the [parts, to sustain them during 
the application of the persulphate of iron, 
and to prevent further bleeding, should it 
occur. The persulphate of iron should be 
kept in or about all working places. 

Bleeding From Scalp Wounds.—A pad or 
compress is placed immediately before the 
ear, over the region marked by a dotted 
line, Fig. 6. The compress is firmly secured 
by a handkerchief. If this does not arrest 
bleeding, a similar compress on the oppo¬ 
site side should be applied. Should the 
bleeding issue from a wound of the posterior or back part of 
the head, a compress should be placed behind the ear, over the 
region marked by the dotted line, Fig. 6, and firmly secured 
by a handkerchief or bandage. 




TREATMENT OF PERSONS OVERCOME 

BY GAS 

Miners are exposed to asphyxia when the circulation of the 
air is not sufficiently active, when the mine exhales a quantity 
of deleterious gas, when they penetrate into old and abandoned 
workings, and when there is an explosion. The symptoms 
of asphyxia are sudden cessation of the respiration, of the 



MINING 


319 


pulsations of the heart, and of the action of the senses; the 
countenance is swollen and marked with reddish spots, the 
eyes are protruded, the features are distorted, and the face is 
often livid, etc. The best and first remedy to employ, and in 
which the greatest confidence ought to be placed, is the renewal 
of the air necessary for respiration. Proceed as follows: 

1. Promptly withdraw the asphyxiated person from the 
deleterious place and expose him to pure air. 

2. Loosen the clothes round the neck and chest, and dash 
cold water in the face and on the chest. 

3. Attempts should be made to irritate the mucous mem¬ 
brane with the feathered end of a quill, which should be gently 
moved in the nostrils of the insensible person, or to stimulate 
it with a bottle of volatile alkali placed under the nose. 

4. Keep up the warmth of the body, and apply mustard 
plasters over the heart and around the ankles. 

5. If these means fail to produce respiration, Docter Syl¬ 
vester’s method of producing artificial respiration should be 
tried as follows: Place the patient on the back on a flat sur¬ 
face, inclined a little upwards from the feet; raise and support 
the head and shoulders on a small firm cushion or folded article 
of dress placed under the shoulder blades. Draw forwards 
the patient’s tongue and keep it projecting beyond the lips; an 
elastic band over the tongue and under the chin will answer 
this purpose, or a piece of string or tape may be tied around 
them, or by raising the lower jaw the teeth may be made to 
retain the tongue in that position. Remove all tight clothing 
from about the neck and chest, especially the suspenders. 
Then standing at the patient’s head, grasp the arms just above 
the elbows, and draw the arms gently and steadily upwards 
above the head, and keep them stretched upwards for 2 sec. 
(by this means air is drawn into the lungs). Then turn down 
the patient’s arms and press them gently and firmly for 2 sec. 
against the sides of the chest (by this means air is pressed 
out of the lungs). Repeat these measures alternately, deliber¬ 
ately, and perseveringly about 15 times in a minute, until a 
spontaneous effort to respire is perceived, immediately upon 
which cease to imitate the movements of breathing, and pro¬ 
ceed to induce circulation and warmth. 


320 


MINING 


6. To promote warmth and circulation, rub the limbs 
upwards with firm, grasping pressure and energy, using hand¬ 
kerchiefs, flannels, etc. Apply hot flannels, bottles of hot 
water, heated bricks, etc., to the pit of the stomach, the arm 
pits, between the thighs, and to the soles of the feet. 

7. On the restoration of life, a teaspoonful of warm water 
should be given, and then, if the power of swallowing has 
returned, small quantities of wine, warm brandy and water, 
or coffee should be administered. 

8. These remedies should be promptly applied, and as 
death does not certainly appear for a long time, they ought 
only to be discontinued when it is clearly confirmed. Absence 
of the pulsation of the heart is not a sure sign of death, neither 
is the want of respiration. 


Promotion 

Advancement in Salary 

and 

r Business Success p 

Secured 
Through the 

COAL MINING 

Mine Foremen’s 
Fire Bosses’ 

Metal Mining 
Metallurgy 
Mining Engineering 

COURSES OF INSTRUCTION 
OF THE 

International 
Correspondence Schools 

International Textbook 
Company, Proprietors 

SCRANTON, PA., U. S. A. 




SEE FOLLOWING PAGES 









General Superintendent 
Over l s 000 Men 

When I enrolled in the Complete Coal Mining 
Course of the International Correspondence 
Schools, my education was confined to a knowl¬ 
edge of how to read and write. Notwithstand¬ 
ing the disadvantage of so poor an education, 
your instruction carried me through and I 
passed a creditable examination. I consider 
that it is to your Schools that I owe my advance¬ 
ment. When I enrolled I was a mine boss; 
I am now General Superintendent for the 
National and Parkdale Fuel Companies, of 
Denver, Colo., and my salary has been in¬ 
creased 125 per cent. There never was a time 
in the history of the United States when good, 
competent, and reliable mine foremen and 
superintendents were so much in demand as 
at present; and any intelligent mine worker 
who has the ambition can fit himself to assume 
the responsibility by taking a Course in Mining 
with the I. C. S. and completing it in leisure 
time that could not be spent in anything more 
advantageous to himself. 

Jos. Watson, 

Louisville, Colo. 


2 



BEGAN WORKING WHEN 8 YEARS OLD 

J. M. Baker, Woodland, Pa., had received but an imper¬ 
fect education when he took up our Complete Coal Mi nin g 
Course, having begun to work in the mine at the age of 8. 
At the time of his enrolment his wages were so small that he 
could with difficulty support himself. His Course has been 
of the greatest benefit to him, enabling him to become mine 
superintendent for the Harbison-Walker Refractories Com¬ 
pany, having eight foremen and several hundred men at work 
under his direction. His wages have been increased 500 per 
cent. 

BECAME GENERAL MANAGER 

Wm. Hollis, Cordova, Ala., says that it was his Complete 
Coal Mining Course with the I. C. S. which enabled him to 
rise from the position of mine foreman to that of general man¬ 
ager of the Alberta Coal, Mineral and Lumber Company. 
His salary has been doubled since he enrolled with us. 

DRAWS $300 A MONTH 

While working as top boss of a small mine, Gus Blair, 
Murphysboro, Ill., enrolled with the I. C. S. for the Complete 
Coal Mining Course. As he had worked in the mines from the 
age of 9, it was hard work for him to confine himself to study. 
However, he pursued the Course until his education was 
improved, and made a substantial advancement in position and 
salary. At the time of enrolment he was paid $50 a month. 
He now draws $300 a month from the Gus Blair Big Muddy 
Coal Company, of which he is half owner and general manager. 

A GRADUATE’S SUCCESS 

S. J. Routledge, Kellerman, Ala., holds I. C. S. Diplomas 
both in Coal Mining and in Surveying and Mapping. When 
he enrolled for the first Course he was earning about $75 a 
month. He is now drawing a salary 150 per cent, larger as 
the superintendent of coal mines for the Central Iron and Coal 
Company. He says that his present position is due solely to 
the knowledge gained from his I. C. S. Course. 

EMPLOYS 500 MEN 

Chas. A. Sine, Johnson City, Ill., left school before he 
knew the multiplication table. When he enrolled for our Coal 
Mining Course, he was driving a mule, earning $40 a month. 
Before he received his diploma he was able to pass the exam¬ 
ination for mine manager. He is now superintendent of the 
Johnson City Coal Company, employing 500 men. 

GRADUATE BECOMES SUPERINTENDENT 

S. B. Isenburg, Osceola Mills, Pa., was working as a laborer 
when he enrolled with the I. C. S. for the Short Coal Mining 
Course, from which he graduated. This enabled him to pass 
the state examination for mine foreman and to become the 
mine superintendent of the Blair Brothers Coal Company, 
with an increase in salary of 150 per cent. 


3 


Income Ten Times 
As Large 

I used to feel that I was working hard enough 
without having to devote my nights to study, 
when employed as a clerk at $45 a month in the 
mining department of the Cambria Steel Com¬ 
pany. However, I stuck to the Complete 
Coal Mining Course, which enabled me to gain 
a first-class certificate of competency as fire 
boss, and afterwards as mine foreman. I am 
at present superintendent of the Johnstown 
mines of the Cambria Steel Company, employ¬ 
ing 2,000 men. In addition, I am a stock¬ 
holder and director in several other companies, 
consequently my income is at least 10 times 
what it was when I enrolled. 

Geo. T. Robinson, 

143 Green St., Johnstown, Pa. 


4 



NOW PROPRIETOR 


James Nevin, Ottumwa, Iowa, while working as a hoisting 
engineer enrolled with the I. C. S. for the Complete Coal 
Mining Course. He gives the Schools the highest indorsement, 
because they have enabled him to become superintendent of 
the Trio Coal Company, of which he is also part owner. 

ONCE A MULE DRIVER 

While driving a mule in the mines at the age of 19. John 
Clapperton, Jr., Minden, W. Va., enrolled for the Complete 
Coal Mining Course. Through the knowledge he obtained 
from this he has been able to pass two examinations and to 
become superintendent of the New River Coal Company, 
largely increasing his salary thereby. 

A GOOD FRIEND OF THE SCHOOLS 

A good friend of the .Schools, Jos. Knapper, Philipsburg, 
Pa., advises his friends to study I. C. S. Courses, because of 
his experience since enrolment for our Complete Coal Mining 
Course. Mr. Knapper was earning $75 a month at the time 
of enrolment. After pursuing hi's Course he rose step by step 
until he is now state mine inspector for the eighth district, at 
a salary of $3,000 a year. 

CREDITS HIS SUCCESS TO THE I. C. S. 

D. J. Griffith, Trinidad, Colo., was earning only $20 a 
month as a miner, at the age of 37, when he enrolled with us 
for the Complete Coal Mining Course. After graduation 
he served the state of Colorado for a time as chief inspector 
of coal mines. He is now chief inspector of mines for the 
American-Victor Fuel Company, and he attributes all his 
success to the I. C. S. 

PAY INCREASED 233 PER CENT. 

P. J. Moore, Carbondale, Pa., was employed as a fire boss 
when he enrolled for the Complete Coal Mining Course. He 
is now state mine inspector for the first anthracite district, 
and pay days bring him 233 per cent, more than they did 
at the time of enrolment. 

DIRECTS 10,000 MEN 

W. R. Calverley, Windber, Pa., was a miner when he en¬ 
rolled for the Complete Coal Mining Course. By diligent 
study he advanced to the position of general superintendent 
of the Berwind-White Coal Mining Company, having the wel¬ 
fare of 10,000 men committed to his charge. He has always 
given great credit to the Schools. 


5 


Now State Mine 
Inspector 

When I enrolled in the Complete Coal 
Mining Course of the International Corre¬ 
spondence Schools, Scranton, Pa., I had had 
about 20 months of schooling all told. I was 
employed at the time as assistant foreman, 
and was getting $55 a month. After enrolling 
in the Schools, I was soon advanced to the 
position of mine foreman at $75 a month, 
which was voluntarily increased to $90 a 
month. I am now Mine Inspector of the Fifth 
Bituminous District, at a salary of $3,000 a 
year. 

Correspondence instruction, as conducted by 
the I. C. S., is the finest and most complete 
in the world today; every young man that 
desires to advance or better his condition should 
enroll at once. No one can enroll with you 
and apply himself to his work, without being 
greatly benefited. 

I shall be glad to answer any inquiries regard¬ 
ing the Schools and my Course with them. 

Isaac G. Roby, 

Inspector, Fifth Bituminous District, Union- 
town, Pa. 


6 



STATE MINE INSPECTOR 

By diligent study of the Complete Coal Mining Course, for 
which he enrolled with the I. C. S., Jos. Williams, 245 Beale 
Ave., Altoona, Pa., has risen from a position as miner to that 
of inspector of mines, for the tenth bituminous district, State 
of Pennsylvania. His salary has increased from $45 a month 
to $3,000 a year, and he gives the I. C. S. all the credit. 

NOW MANAGER 

T. E. Moore, Eyremore, Alta., Can., was working as a 
shift man for the Prairie Coal Company, when he enrolled 
with the Schools for the Complete Coal Mining Course. He 
is still working for the same company, and he has risen to the 
position of manager of their mine on the Bow River at a salary 
of $150 a month. 

COULD HARDLY WRITE HIS NAME 

A. W. Courtney, Princeton, B. C., was 25 years old and was 
working as a laborer when he enrolled for the Short Coal 
Mining Course. At the time he could hardly write his name. 
Keeping diligently at his studies, he was able to pass the exam¬ 
ination for mine foreman and now holds a foreman’s position 
at a salary of $150 a month. 

THREE TIMES HIS FORMER SALARY 

J. J. Clark, Sagamore, Pa., began to reap the benefits of his 
study on the Complete Coal Mining Course 10 months after 
enrolment. By devoting all his spare time to his Course, he 
was able to obtain a mine foreman’s certificate. He is now 
assistant superintendent for the Buffalo & Susquehanna Coal 
and Coke Company, and his wages are three times as great 
as when he was loading coal. 

IN CHARGE OF A LARGE PLANT 

When H. L. Fisher, Kayford, W. Va., enrolled with the 
I. C. S. for the Complete Coal Mining Course his knowledge 
of mining was very limited. By diligent study of his Course 
he is now able to take charge of the largest plant of the Cabin 
Creek Consolidated Coal Company, which includes eight of 
their principal mines. His salary is $190 a month. 

NOW SERVES THE STATE 

F. J. Pearce, Rm. 120, State Capitol, Indianapolis, Ind., 
enrolled with the I. C. S. 12 years ago for the Complete Coal 
Mining Course. He has now risen to the highest position in 
his profession, deputy inspector of mines and mining for the 
State of Indiana, at a salary of $2,000 a year. He had little 
education before enrolment, but the secret of his advancement 
lies in the fact that he has used his spare time and his I. C. S. 
Course to obtain an education. 


7 


$540 to $3,000 a Year 

In reply to your letter, I beg leave to state 
that my*present position is Mine Inspector, 
employed by the State of Pennsylvania. When 
I enrolled in the Full Mining (now the Mining 
Engineering) Course, my salary was $45 a 
month. I was employed as bratticeman, at 
the Woodward Mines. After studying some 
time, I passed the examination for mine fore¬ 
man and was appointed assistant mine fore¬ 
man—later foreman. Then I passed the Mine 
Inspector’s examination and was elected to that 
position. My present salary is $250 a month. 
I can conscientiously recommend the Interna¬ 
tional Correspondence Schools to any young 
man that has any desire to advance himself. 

L. M. Evans, 

Inspector Second Anthracite Inspection Dis¬ 
trict, 10 Belmont Terrace, Scranton, Pa. 


8 



NOW A MINE OWNER 


J. P. Davis, Columbia, Mo., was earning $75 a month as 
mine foreman when he enrolled with the I. C. S. for the Com¬ 
plete Coal Mining Course. This has been so profitable to him 
that he is now senior partner and manager of the Davis & 
Watson Coal Company, employing 30 men. 

NOW A FOREMAN 

Howell John, Box 48, Meritt, B. C., declares that his 
I. C. S. Mine Foremen’s Course advanced him to the position 
of foreman with the Pacific Coast Collieries, Mr. John began 
working in the mines at 13 years of age and was a miner 
when he enrolled. His present position pays him $135 a 
month 

EARNS $200 A MONTH 

While working as a timberman, John Prentice, Lund- 
breck, Alta., Can., took up the Mine Foremen’s Course with 
the I. C. S. At the time he was earning $70 a month. He 
is now mine manager for the Breckenridge & Lund Coal Com¬ 
pany, earning $200 a month, and he says it was the I. C. S. 
that made the difference. 

WORTH $40 A MONTH TO HIM 

His Mine Foremen’s Course with the I. C. S., for which 
Addison Shaw, Berryburg, W. Va., subscribed, was the means 
of advancing him to the position of mine foreman for the Con¬ 
solidated Coal Company, with an increase in salary of $40 a 
month. 


NOW AN OFFICER OF THE COMPANY 

R. S. Burchinal, Smithfield, Pa., says that the Mine Fore¬ 
men’s Course for which he enrolled with the I. C. S., was the 
cause of his obtaining a foreman’s certificate which enables him 
to look after the mine, as well as the outside management of 
his company. He is now treasurer and general manager of the 
Smithfield Coal and Coke Company, receiving a salary of $125 
a month. 


A WORLD OF GOOD 

The experience of G(is Champ, Cherokee, Kans., shows what 
the I. C. S. Mine Foremen’s Course will do for the man that 
has the grit to go ahead. Mr. Champ says that his Course 
did him a world of good, since it has advanced him to the posi¬ 
tion of foreman for the Hamilton Coal Company, at a salary 
of $100 a month. 


9 


Earning $3,000 a Year 

Howard M. Black, 

Mining Engineer, Grass Valley, Cal. 

International Correspondence Schools, 
Scranton, Pa. 

Gentlemen: —At the time of enrolment as 
a student of the I. C. S., I was superintending 
a small mine at a salary of $100 a month. I 
have finished my Metal Mining Course, with 
the exception of geometrical drawing. I still 
hold the same position, but as this only occu¬ 
pies part of my time, I make a specialty of 
examining and reporting on mines, and do con¬ 
siderable assaying and other work in the line 
of mining engineering. For this outside work 
I receive $12 a day and expenses. I can safely 
say that my income since enrolment has been 
increased from the original $1,200 a year to 
$3,000 a year, due in great part to the technical 
knowledge acquired through the I. C. S. Course. 

Very truly yours, 

Howard M. Black, 

Grass Valley. Cal. 


10 




NOW PRESIDENT 


G. W. Wilmott, Maryd, Pa., was earning about $60 a month 
as chief repairman about the mines, when he enrolled for the 
Mine Mechanical Course. After rising to the position of 
superintendent, in charge of 450 men, he resigned to become 
president and general manager of the Wilmott Engineering 
Company, his present position. 

LARGEST OF ITS KIND 

E. E. Carter, Quartzburg, Idaho, commenced studying the 
Complete Metal Mining Course soon after leaving grammar 
school, while working for $10 a week. Later he enrolled for the 
Complete Metallurgy Course. He considers that his success 
is largely due to the instruction he received from the I. C. S. 
At present he is manager of the largest coal mines in Idaho. 

FROM $2.50 A DAY TO $38.20 A WEEK 

Henry Hoard, Selwood, Ore., feels that he owes to the 
I. C. S. all his success. He was working around the mines as 
a mucker, or at anything else he could get, when he enrolled 
for the Metal Mining Course. His wages then averaged $2.50 
a day. He is now assistant foreman for the John Clark Lead 
Company, employing some 40 men, and his salary averages 
$38.20 a week. 


BECAME SUPERINTENDENT 

E. A. Roberts, Entwistle, Alta., Can., was working as a 
steam engineer for $80 a month when he enrolled with the 
Schools for the Mining Engineering Course. He is now man¬ 
ager of the shaft sinking work for the Pembina Coal Company, 
at a salary of $150 a month. 

EARNS $160 A MONTH 

Geo. H. Shepherd, National, Nev., was earning $2 a day 
as a millman and sampler, when he enrolled with the I. C. S. 
for the Metal Mining Course. He now handles the retorting 
of the huge mass of bullion dispatched from the camp, earning 
$160 a month. 

WHAT THE SCHOOLS DID FOR HIM 

In 1909, Peter Kasavage, Johnson City, Ill., could neither 
read nor write and was earning $2.42 a day as a tracklayer. 
He then enrolled with the I. C. S. for the Mine Foremen’s 
Course. In July, 1911, he passed the state examination for 
mine examiner and the next day was appointed to the position 
of mine examiner of the Illinois Hocking Washed Coal .Com¬ 
pany, at Marion, Ill. 


11 


A Young Man’s Success 

When I began with the Desoto Coal Mining 
Development Company I counted cap boards. 
The president advised me to study Mining 
Engineering through your Schools. As a result 
of this study promotion and advancement 
have been my lot, together with commensu¬ 
rate compensation. My salary has increased 
400 per cent, since taking up instruction by 
mail. If one received the mental training 
alone, the Course would be worth many times 
its cost. Today I am Secretary and General 
Manager of the company and a director and 
stockholder. To have gone step by step from 
a counter of cap boards, to the Secretary and 
General Manager’s chair has meant hours and 
nights and weeks and months of study as well 
as close application to my duty. 

Jas. A. Worsham, 

Morris, Ala. 


12 



SALARY DOUBLED 


Nels Johnson, Zeigler, Ill., was earning $60 a month when 
he enrolled with the I. C. S. for the Short Coal Mining Course. 
He afterward graduated from the Full Mining Course. He is 
now mine manager of the Bell & Zoller Mining Company’s 
plant, and his salary has been doubled. 

EIGHT TIMES HIS FORMER SALARY 

Henry Sankey, Box 756, Cobalt, Ont., was working on a 
farm at $20 a month when he took up our Metal Mining Course. 
This has enabled him to become superintendent of the Peter¬ 
son Lake Mining Company, at a salary of $160 a month. 

SIX TIMES HIS FORMER SALARY 

E. F. Buffat, Briceville, Tenn., was supporting a family 
on the small salary of a bookkeeper when he enrolled with 
the I. C. S. for the Metal Mining Course. He is now super¬ 
intendent of the Tennessee Coal Company’s mine at Brice¬ 
ville, employing about 200 men, at a salary six times what he 
received at the time of enrolment. 

EARNS $250 A MONTH 

Lewis R. Smith, 314 Second Ave., Rome, Ga., was an assist¬ 
ant chemist, earning $45 a month when he enrolled with the 
Schools. His Course in Metallurgy has enabled him to be¬ 
come superintendent of the Silver Creek Furnace Company 
at a salary of $250 a month. 

NOW SUPERINTENDENT 

J. W. Powell. Tabei, Alta., was working as a fire boss, 
when he enrolled with the Schools for the Complete Coal 
Mining Course. Through the study of this Course he qualified 
himself for a first-class certificate in the Pennsylvania anthra¬ 
cite region, and afterward for the mine foreman’s position in 
Alberta. He is now superintendent of mines for the Canada 
West Coal Company, Ltd. 

WORKING AGAINST ODDS 

D. R. Jones, Parrot, Va., had attended school only a few 
months when he enrolled for our Complete Coal Mining Course. 
In spite of obstacles he has advanced step by step until he is now 
superintendent of the Pulaski Anthracite Coal Company, at a 
salary of $150 a month. 


13 


Increased His Salary 
500 Per Cent. 

I had learned the machinist’s trade and was 
prepared to enter Cornell University, when I 
enrolled for your Coal Mining Course. I had 
no knowledge of mine engineering or metal 
mining at the time, but had worked in the coal 
mines. In making the radical change from the 
drift coal mines to the comparatively deep 
lead mines of this section, I found that my 
knowledge of metal mining was thorough. 
I was able to make my work very successful 
from the start. I found my Course very use¬ 
ful in all my work, and have always been a 
firm advocate of the I. C. S. At the time of 
enrolment I was earning $600 a year. I am 
now general superintendent of the Washburn 
Lignite Coal Company, of this place, employ¬ 
ing 300 men, and my salary has increased 500 
per cent. 

A. W. Pollock, 

Wilton, N. Dak. 


14 



NOW MANAGER 

Robt. Elminston, Box 696, Panama, Ill., had to start life 
with a common school education; but with the help of our 
Short Coal Mining Course, he was able to obtain a first-class 
certificate in the State of Illinois. This advanced him from a 
position as miner to that of manager of the Shoal Creek Coal 
Company’s mine No. 1. 

NOW SUPERINTENDENT 

Geo. E. Loughner, R. F. D. No. 4, Johnstown, Pa., went 
into the mines at 11 years of age to help support a large family. 
While he was earning about $50 a month he enrolled for our 
Short Coal Mining Course, and afterwards for the Complete 
Coal Mining Course. He is superintendent for the Kelso 
Smokeless Coal Company, employing 125 men, and his salary 
has been increased $75 a month. 

PASSED WITH 100 PER CENT. 

While John Sanderson, Red Lodge, Mont., was working 
as a miner, he enrolled with the I. C. S. for the Short Coal 
Mining Course. When he came up for examination he was 
able to pass with a percentage of 100, although his previous 
education had been greatly limited, owing to the fact that he 
began to work in the mines at 10 years of age. He is now 
acting as foreman for the Northwestern Improvement Com¬ 
pany, and his salary has increased 125 per cent. 

BECAME FOREMAN 

Wm. Fleming, Windber, Pa., was working in the coal mines 
when he enro led with the I. C. S. for the Short Coal Mining 
Course. What he learned enabled him to obtain a first-grade 
mine foreman’s certificate, and he now holds the position as 
foreman of the Eureka No. 42 Mine of the Berwind-White 
Coal Mining Company. His salary has been increased 60 per 
cent, since enrolment. 

INCREASED HIS SALARY 

Because he had studied the Short Coal Mining Course for 
which he enrolled with the I. C. S., John H. Hauser, Mar¬ 
guerite, Pa., was able to advance from the position of driver 
to that of mine foreman, increasing his salary to $135 a month. 

MINER BECAME SUPERINTENDENT 

J. C. Glancy, Pineville, Ky., was working as a miner when 
he subscribed for the Mine Foremen’s Course.. Two years 
ago he secured a state certificate and is now holding a position 
as mine superintendent for the Pioneer Coal Company, at a 
salary of $100 a month. He says that the I. C. S. have done 
wonderful things for him. 


15 


State Mine Inspector 
Salary $3,000 

Since I had started to work in the mines 
when only 9 years old, my education at the 
time when I enrolled with the International 
Correspondence Schools for the Short Coal 
Mining Course, was principally what I had 
picked up by observation in the school of 
experience. Without question, your Course 
offered the best advantage I had ever had, 
and the farther I went with it the better I 
liked it. I was so pleased with your treatment 
of your scholars that I have recommended the 
I. C. S. to other miners who are now holding 
positions as foremen and assistant foremen 
in my district, which should be sufficient, guar¬ 
antee that I have every faith in the I. C. S. 
My work with the Schools was my best prep¬ 
aration to stand for an examination for State 
Inspector of Mines, which, I am pleased to say, 
I was successful in passing. I am now Inspec¬ 
tor for the Seventeenth Anthracite Inspection 
District. When I took up your Course I was 
receiving a salary of $60 a month. My present 
salary is $3,000 a year; therefore, you see I have 
every reason to praise the bridge that carried 
me over. 

I recommend your Course to any one, young 
or old, who has ambition. It matters not what 
his previous schooling has been, the I. C. S. 
will see him through. 

Isaac M. Davies, 

Lansford, Pa. 


16 








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